Μιχάλης Παπαδημητράκης Ανάλυση Πραγματικές Συναρτήσεις μιας Μεταβλητής Τμήμα Μαθηματικών Πανεπιστήμιο Κρήτης
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. () , , , . , , . .
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, , , . , , Bolzano - Weierstrass , , .
; - . , , , - , , . , , , . .
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1. , , - - : - , , . . , , . , supremum: - .
2. . , n n 2.2, 2.3. , 3.5, 3.6.
3. , , 10 . , , - .
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4. Riemann , , Darboux . , Riemann . Riemann , Darboux - Darboux Riemann.
5. . , -. , (, ), .
6. , , . , ( , Riemann , , ) . : . .
7. Peano. , , . Foundations of Analysis E.Landau . () () Dedekind. , , Cauchy , ., , : , .
8. . , Cauchy .
9. , , , , . : - , , - . , , .
, , :
Mathematical Analysis, T. Apostol.
Dierential and Integral Calculus, R. Courant.
The Theory of Functions of Real Variables, L. Graves.
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Foundations of Analysis, E. Landau.
Principles of Mathematical Analysis, W. Rudin.
A Course of Higher Mathematics, V. Smirnov.
Calculus, M. Spivak.
The Theory of Functions, E. C. Titchmarsh.
, ,
Calculus, T. Apostol.
Introduction to Calculus and Analysis, R. Courant - F. John.
, ,
, . (http://users.uoa.gr/apgiannop/). , . - . - . .
, , ., , , .
2011.
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I . 1
1 . 31.1 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Supremum inmum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 . 212.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 , , . . . . . . . . . . . 212.1.2 n 2 N . n 2 N. . . . . . 23
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 . e, . . . . . . . . . . . . . . . . . . . . . . . . . 432.6 Supremum, inmum . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.9 limsup liminf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 . 703.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1.2 . . . . . . . . . . . . . 74
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.2.3 , , . . . . . . . . . . . . . . . . . . . . . 86
3.3 . . . . . . . . . . . . . . . . . . . . . . . . 883.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
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3.5.3 . . . . . . . . . . . . . . . . . . . . . 1033.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.7 Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4 . 1124.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5 . 1515.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.1.3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.6 . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.6.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.7 . . . . . . . . . . . . . . . 1855.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.7.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1875.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1905.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.8 . . . . . . . . . . . . . . . . . . . . . . . . . 1995.8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1995.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.9 , . . . . . . . . . . . . . . . . . . . . . . . . . 2065.9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2065.9.2 . . . . . . . . . . . 207
5.10 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.12 . . . . . . . . . . . . . . . . . . . . . . . . . 215
6 Riemann. 2216.1 Darboux. . . . . . . . . . . . . . . . . . . . . . . . . . 2216.2 . Darboux. . . . . . . . . . . . . . . . . . . . . . . . . 224
6.2.1 . . . . . . . . . . . . . . . . . . . . . . 2246.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
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6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2326.5 . Riemann. . . . . . . . . . . . . . . . . . . . . . . . . 249
7 . 2577.1 , . . . . . . . . . . . . . . . . . . . . . . . . 257
7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2627.3 . . . . . . . . . . . . . . . . . . . . . . . . . 269
7.3.1 . . . . . . . . . . . . . . . 2697.3.2 . . . . . . . . . . . . 2707.3.3 . . . . . . . . . . . . . . . . . . . . . . 2727.3.4 . . . . . . . . . . . . . . . . 2757.3.5 . . . . . . . . . . . . . . 280
7.4 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.5 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . 293
7.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2957.5.4 Simpson. . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2967.6.1 . . . . . . . . 2977.6.2 . . . . . . . . . . . . . . . . . . . . . . 2997.6.3 . . . . . . . . . . . . . . . . . . . . . . 301
7.7 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . 3067.7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3067.7.2 . . . . . . . . . . . . . . . . . . . . . . 3077.7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3107.7.4 . . . . . . . . . . . . . . . . . . . . . . . . . 3127.7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
8 . 3188.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3188.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3238.3 p- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3308.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
8.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3348.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3448.5.1 . . . . . . . . . . . . . . . . . . . . . . . 3448.5.2 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3488.5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
9 . 3559.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3559.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3579.3 Weierstrass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
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10 . 37110.1 . . . . . . . . . . . . . . . . . . . . . . . 37110.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38010.3 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39210.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
10.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 39810.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 400
11 . 40411.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40411.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41011.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
11.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41411.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
11.4 . . . . . . . . . . . . . . . . . . . . . . . 42011.4.1 . . . . . . . . . . . . . . . . . . . . . 42011.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 42311.4.3 . . . . . . . . . . . . . . . . . . . 424
11.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
II . 440
12 . 44212.1 Peano. . . . . . . . . . . . . . . . . . . . . . . . 442
12.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44212.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44512.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44812.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44812.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44912.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45112.2.4 . . . . . . . . . . . . . . . . . . . . . . . 453
12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45412.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45512.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45612.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45812.3.4 R+ . . . . . . . . . . . . . . . . . . . . . . . . . 46012.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . 461
12.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46412.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46412.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46512.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46612.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46712.4.5 R. . . . . . . . . . . . . . . . . . . . . . . . . . 467
12.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46812.5.1 Cauchy. . . . . . . . . . . . . . . . . . . . . . 46812.5.2 . . . . . . . . . . . . . . . . . . 469
12.6 : R N. . . . . . . . . . . . . . . . . . . . . 470
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12.6.1 R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47012.6.2 R. . . . . . . . . . . . . . . . . . . . . . . . . . 47112.6.3 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
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I
.
1
-
2
-
1
.
1.1 R . R. ,
, , N, Z Q. R, .
R , . .
. R = R [ f1;+1g R. +1, 1 , .
R.
. , 1 < x x < +1 x 2 R 1 < +1. (+1) = 1 (1) = +1. (+1)+x = +1, x+(+1) = +1, (1)+x = 1, x+(1) = 1 x 2 R (+1) + (+1) = +1, (1) + (1) = 1., (+1) + (1), (1) + (+1) - . (+1) x = +1, x (1) = +1, (1) x = 1, x (+1) = 1 x 2 R (+1) (1) = +1, (1) (+1) = 1. (+1)(+1), (1)(1) . (1)x = 1 x(1) = 1 x > 0, (1)x = 1 x(1) = 1 x < 0 (1)(1) = +1, (1)(1) = 1. (1)0, 0(1) . 1
+1 = 0,11 = 0.
10 .
1x
= 1 x > 0, 1x
= 1 x < 0 x1 = 0 x. x
0, 1
0, 11 ,
11 .
, j+1j = +1, j 1j = +1.
( ) ( ) . , : (+1) + (+1) = +1. ,
3
-
, : (+1) (+1) . , ( ) , , : (+1)0 , , . .
. 1. , , - (a; u; x;m; n; ; " ) (M;S ) , R. , , , , R. 1 1, . : a 2 R, 2 (a;+1], A R, A [1; 3].2. N = f1; 2; 3; : : : g. , 0 .
.
1. - ( ) 1 ( ) ( ) .
2. R R . , (, , ) R R , . : (xy)z = x(yz) x; y; z 2 R , xy, yz, (xy)z, x(yz).
3. x y < 0 z w < 0, 0 < yw xz.4. [] x y, z w, t s x+ z + t = y + w + s, x = y, z = w, t = s.
[] 0 < x y, 0 < z w, 0 < t s xzt = yws, x = y, z = w, t = s.5. .
[] jxj a a x a.[] jxj < a a < x < a.[] :
jjxj jyjj jx yj jxj+ jyj:
[] jx+ yj = jxj+ jyj x; y 0 x; y 0.[] jx+y+zj jxj+ jyj+ jzj. , jx+y+zj = jxj+ jyj+ jzj x; y; z 0 x; y; z 0.
6. .
a x b a y b, jx yj b a.
4
-
7. .
[] b1; : : : ; bn > 0 a1b1; : : : ; an
bn< u, a1++an
b1++bn < u.
[] b1; : : : ; bn > 0 a1b1; : : : ; an
bn> l, a1++an
b1++bn > l.
8. [] x : jx+1j > 2, jx 1j < jx+1j,x
x+2> x+3
3x+1, (x 2)2 4, jx2 7xj > x2 7x, (x1)(x+4)
(x7)(x+5) > 0,(x1)(x3)
(x2)2 0.[] x x : (1; 3], (2;+1),(3; 7), (1;2) [ (1; 4) [ (7;+1), [2; 4] [ [6;+1), [1; 4) [ (4; 8], (1;2] [ [1; 4) [[7;+1).
1.2 .
R, . , , : , , . , A;B - A B, : , .
. - A;B a b a 2 A, b 2 B. a b a 2 A, b 2 B.
R . , 1.5 Q .
1.1. b n 2 N n > b.. - - b n b n 2 N. B = fb jn b n 2 Ng ., n b n 2 N, b 2 B. , n b n 2 N, b 2 B. 1 < , 1 B. n 2 N n > 1 , , n+ 1 > . , n+ 1 2 N. . a > 0 n 2 N 1
n< a .
. 1.1 b = 1a.
1.1. x k 2 Z k x < k + 1.. 1.1 n 2 N n > x m 2 N m > x. l = m, l; n l < x < n. , k 2 Z, k x k + 1 x. l x, k x k 2 Z, k l. , , n > l n > x. , , k 2 Z k x k + 1 > x, k x < k + 1. k k x < k+1 . k x < k+1 k0 x < k0+1 k; k0 2 Z. k < k0+1 k0 < k+1, 1 < k0k < 1. k0 k 2 Z, k0 k = 0, k0 = k.
5
-
. k 2 Z k x < k + 1, 1.1, x [x].
, [x] x < [x] + 1 [x] 2 Z. . a; b, a < b r 2 Q a < r < b.. 1.1, n 2 N n > 1
ba . na < [na] + 1 na+1 < nb. m = [na] + 1, a < m
n< b,
r = mn
a < r < b.
.
1. . 6 .
[] a " " > 0, a 0.: a > 0. , a, " > 0 " < a.
[] a b+ " " > 0, a b.[] ja bj " " > 0, a = b.[] x a x < b, b a. x a x > b, b a.
2. [a; b) (a;+1) .
3. A = (1; 0]; B = [0;+1) : a b a 2 A, b 2 B. A = (1; 0], B = (0;+1), A = (4;2), B = (2;+1) A = (1; 0), B = [1; 13].
4. 2.9.
- A;B A [ B = R a < b a 2 A, b 2 B. A = (1; ); B = [;+1) A = (1; ]; B = (;+1).
5. [] - A;B a b a 2 A, b 2 B " > 0 a 2 A, b 2 B b a ". a b a 2 A, b 2 B.[] - A;B 0 < a b a 2 A, b 2 B " > 0 a 2 A, b 2 B b
a 1 + ".
a b a 2 A, b 2 B.6. [] a 1
n n 2 N, a 0.
[] a b+ 1n n 2 N, a b.
[] ja bj 1n n 2 N, a = b.
1: 1 R .
7. A = f 1njn 2 Ng, B = f 1
njn 2 Ng
. a b a 2 A, b 2 B. A = fr 2 Q j r < 0g, B = fr 2 Q j r > 0g.
6
-
8. .
[] r a r 2 Q, r < b, b a.[] fr 2 Q j r > ag = fr 2 Q j r > bg, a = b.[] fr 2 Q j r < ag \ fr 2 Q j r > bg = ;, a b.[] fr 2 Q j r ag [ fr 2 Q j r bg = Q, b a.
9. [] , , . 6= 0 .[] , , .
10. .
[b]: n 2 Z n > b, n 2 Z n b, n 2 N n > b n 2 N n b. a > 0, [ 1
a] n 2 N 1
n< a.
11. [] k 2 Z, [x+ k] = [x] + k.[] [x+y] = [x]+ [y] [x+y] = [x]+ [y]+1. [x+ y + z].
[] [x] + [x+ 1n] + + [x+ n2
n] + [x+ n1
n] = [nx] n 2 N, n 2.
12. k 2 N, a < b. ba, r 2 (a; b) r = m
n, m;n 2 Z, 1 n k.
1.3 .
1.3.1 .
. , n 2 N, a n an = a a, n a., a 6= 0, n 2 Z, n < 0, a0 = 1 an = 1
an =1
aa , jnj = n a. 00 .
, n 2 Z , (a)n = an , n 2 Z , (a)n = an . , n 2 Z , an > 0 a 6= 0. n 2 Z , an > 0, a > 0, an < 0, a < 0.
1.2 . .
1.2. 1. :axbx = (ab)x , axay = ax+y , (ax)y = (ay)x = axy .2. 0 < a < b, (i) ax < bx , x > 0, (ii) a0 = b0 = 1, (iii) ax > bx , x < 0.3. x < y, (i) ax < ay , a > 1, (ii) 1x = 1y = 1, (iii) ax > ay , 0 < a < 1.
. 1. : x > 0, axbx = (a a)(b b) = (ab) (ab) = (ab)x . x > 0. : x; y > 0, axay = (a a)(a a) = a a = ax+y . x; y > 0.
7
-
: x; y > 0, (ax)y = ax ax = (a a) (a a) = a a = axy . x; y > 0 (ay)x = axy (ax)y = axy x; y.2. (i) a < b x , ax = a a 0. 0 n = a, , , > 0. Y = fy j y > 0; yn ag Z = fz j z > 0; zn ag. y0 = minfa; 1g z0 = maxfa; 1g. y0 2 Y z0 2 Z , Y; Z . y 2 Y , z 2 Z yn a zn, yn zn , y; z > 0, y z. , y z y 2 Y , z 2 Z. n = a. 0 < " < . " < , " Z. " 0 ( ")n < a. , ( ")n < a. Bernoulli, a
n>1 "
n 1 n ",
na
nn1 < ". " > 0, " < , " > 0. nann1 0,
n a. " > 0. +" > , +" Y . +" 0 (+")n > a. , (+")n > a. Bernoulli,
n
a>
+"
n=1 "
+"
n 1n "+"
,
an
na< "
+"< "
, , (a
n)na
< ". " > 0, (an)
na 0
, , n a. n a n a n = a. xn = a. 1; 2 > 0, 1
n = a 2
n = a, 1n = 2
n , 1 = 2 .
8
-
1.2 4.4 7 2.5.
. n 2 N, a 0, - xn = a, 1.2, n- a n
pa .
, np0 = 0 n
pa > 0, a > 0.
1.2 - xn = a. 1.3 ; , 1.2, .
1.3. 1. n 2 N , xn = a : (i) , npa npa , a > 0, (ii) , np0 = 0, a = 0, (iii) ,
a < 0.2. n 2 N , xn = a : (i) npa , a > 0,(ii) n
p0 = 0, a = 0, (iii) npa , a < 0.
- - . .
1.4. n; k. npk k n-
. , npk , .
. k n- , m 2 N k = mn . n
pk = m , , .
, npk . n
pk = m
l, m; l 2 N -
- m; l > 1. k = m
n
ln, lnk = mn . ,
> 1 , . , l lnk, mn = mn1m. l > 1 m, l mn1 = mn2m. , l mn2 = mn3m. , l m0 = 1. l = 1, k = mn k n- .
, , R . 1.2 , , .
1.5. R nQ .. 2 , 1.4
p2
.
, , , Q : 2. 1.2 Q. , x2 = 2 Q.
. a; b, a < b x 2 R nQ a < x < b.. c. a c < b c, r 2 Q a c < r < b c. r + c a < r + c < b.
9
-
1.3.3 .
1.1. a > 0, m; k 2 Z, n; l 2 N mn= k
l. ( n
pa)m = ( l
pa)k .
. (( npa)m)nl = (( n
pa)n)ml = aml (( l
pa)k)nl = (( l
pa)l)kn = akn .
ml = kn, aml = akn , (( npa)m)nl = (( l
pa)k)nl . , ( n
pa)m > 0
( lpa)k > 0, ( n
pa)m = ( l
pa)k .
. , a > 0, r 2 Q. m 2 Z, n 2 N r = mn.
, 1.1, ( npa)m . ,
ar = ( npa)m .
, r 2 Q, r > 0, 0r = 0. , 0r , 0r = ( n
p0)m r = m
n, m;n 2 N. ,
a1n = n
pa , a 0, n 2 N. , ar ,
ar > 0 a > 0, r 2 Q. 1.2 . . x = m
n y = k
l, m; k 2 Z, n; l 2 N.
, (ax)n = (( npa)m)n = (( n
pa)n)m = am ,
(ax)n = am :1. : a; b > 0. (axbx)n = (ax)n(bx)n = ambm = (ab)m = ((ab)x)n , axbx > 0 (ab)x > 0, axbx = (ab)x . a = 0 b = 0 . : x+y = ml+kn
nl. a > 0, (axay)nl = (ax)nl(ay)nl = ((ax)n)l((ay)l)n =
(am)l(ak)n = amlakn = aml+kn = (ax+y)nl , axay > 0 ax+y > 0, axay = ax+y . a = 0 . : xy = mk
nl. a > 0. ((ax)y)nl = (((ax)y)l)n = ((ax)k)n = ((ax)n)k =
(am)k = amk = (axy)nl , (ax)y > 0 axy > 0, (ax)y = axy . (ay)x = axy
(ax)y = axy x; y. a = 0 .2. (i) x > 0, m > 0. (ax)n = am < bm = (bx)n , ax > 0 bx > 0, ax < bx . (iii) (i) (ii) .3. (i) (ax)nl = ((ax)n)l = (am)l = aml (ay)nl = ((ay)l)n = (ak)n = akn . x < y n; l > 0, ml < kn. aml < akn , (ax)nl < (ay)nl . ax > 0 ay > 0, ax < ay . (iii) (i) (ii) .
1.3.4 .
a > 1. r; s; t 2 Q, s < r < t as < ar < at . , s; t 2 Q, x 2 R n Q, s < x < t as < ax < at . as ; at ax . , ax : as < ax < at s; t 2 Q, s < x < t. 1.2. a > 1.1. b > 1 n 2 N bn > a.2. b a 1n n 2 N, b 1.
10
-
. 1. n 2 N n > a1b1 . Bernoulli,
bn = (1 + b 1)n 1 + n(b 1) > a.2. 1.
1.3. 1. a > 1, x 2 R n Q. as < < at s; t 2 Q, s < x < t.2. a > 1, x 2 Q. as < < at s; t 2 Q, s < x < t ax .
. 1. x 2 R nQ. S = fas j s 2 Q; s < xg T = fat j t 2 Q; t > xg. S; T , s; t 2 Q s < x < t. , S T . , s < x < t s < t , a > 1, s; t 2 Q, as < at . S; T , as at s; t 2 Q, s < x < t. as < < at . , , s0; t0 2 Q s < s0 < x < t0 < t, as < as0 at
0< at , , as < < at s; t 2 Q, s < x < t.
2. x 2 Q. = ax , as < < at s; t 2 Q, s < x < t. , 1 2, . 1; 2 a
s < 1 < at as < 2 < a
t s; t 2 Q, s < x < t. 2
1< ats 1
2< ats s; t 2 Q, s < x < t.
, n 2 N s; t 2 Q x 12n< s < x < t < x + 1
2n.
t s < 1n, , 2
1< a
1n 1
2< a
1n .
n 2 N, 1.2 21 1 1
2 1, 1 = 2 .
. a > 1, x 2 R n Q, ax 1.3.
, , ax as < ax < at s; t 2 Q,s < x < t . 1.3, a > 1, x 2 Q ax .
9 1.5 ax , a > 1 x 2 R nQ.. a = 1, x 2 R nQ, 1x = 1. 0 < a < 1, x 2 R nQ, 1
a> 1, ( 1
a)x . ax = 1/( 1
a)x .
, x 2 R nQ, x > 0, 0x = 0., , ax -
: (i) a > 0, (ii) a = 0, x > 0 (iii) a < 0, x 2 Z. , ax : (i) a = 0, x 0 (ii) a < 0, x 2 R n Z.
. 4.3 - ax a < 0 x .
ax . a > 0. x 2 Q, ax > 0. x 2 R n Q, , ax , ax > as s 2 Q, s < x,, as > 0, ax > 0. ax > 0 a > 0 x.
11
-
1.3. 1. s; t 2 Q, s < x + y < t. s0; s00; t0; t00 2 Q s0 < x < t0,s00 < y < t00 , s = s0 + s00 , t = t0 + t00.2. x; y > 0, s; t 2 Q, 0 < s < xy < t. s0; s00; t0; t00 2 Q 0 < s0 < x < t0,0 < s00 < y < t00 , s = s0s00 , t = t0t00.
. 1. , s0 2 Q s y < s0 < x s00 = s s0 2 Q. s0 < x, s = s0 + s00 s00 = s s0 < y. , t0 2 Q x < t0 < t y t00 = t t0 2 Q. x < t0, t = t0 + t00 y < t t0 = t00.2. s0 2 Q s
y< s0 < x s00 = s
s0 2 Q. s0 < x, s = s0s00 s00 = s
s0 < y. , t0 2 Q x < t0 < t
y t00 = t
t0 2 Q. x < t0,t = t0t00 y < t
t0 = t00.
1.2 . - .1. : a; b > 1. s; t 2 Q, s < x < t as < ax < at bs < bx < bt , (ab)s = asbs < axbx < atbt = (ab)t . axbx (ab)s , (ab)t s; t 2 Q, s < x < t, (ab)x axbx = (ab)x . - - a; b > 1. : a > 1. s; t 2 Q, s < x+ y < t , 1.3, s0; s00; t0; t00 2 Q s0 < x < t0, s00 < y < t00 , s = s0 + s00 , t = t0 + t00 . as0 < ax < at0 as
00< ay < at
00, , as = as
0as
00< axay < at
0at
00= at . axay
as , at s; t 2 Q, s < x+y < t, ax+y axay = ax+y . a > 1. : a > 1, x; y > 0. s; t 2 Q, s < xy < t s1 2 Q s1 s, 0 < s1 < xy < t. 1.3, s0; s00; t0; t00 2 Q 0 < s0 < x < t0, 0 < s00 < y < t00 , s1 = s0s00 , t = t0t00. 1 < as
0< ax < at
0, ,
as as1 = (as0)s00 < (ax)s00 < (ax)y < (ax)t00 < (at0)t00 = at . (ax)y as , at s; t 2 Q, s < xy < t, axy (ax)y = axy . a > 1, x; y > 0 (ay)x = axy (ax)y = axy x; y.2. (i) s 2 Q 0 < s < x, , 1 < b
a, 1 < ( b
a)s < ( b
a)x .
, ax < ax( ba)x = (a b
a)x = bx . (ii) (iii) (i).
3. (i) r 2 Q x < r < y, ax < ar < ay . (ii) (iii) (i).
2 1.2 ax , a, : (i) (0;+1), x > 0, (ii) 1 (0;+1), x = 0, (iii) (0;+1), x < 0.
3 1.2 ax , x, : (i) (1;+1), a > 1, (ii) 1 (1;+1), a = 1, (iii) (1;+1), 0 < a < 1.. a+1 = 0, 0 a < 1, a+1 = +1, a > 1. a1 = +1, 0 < a < 1, a1 = 0, a > 1. (+1)b = +1, b > 0 b = +1, (+1)b = 0 b < 0 b = 1. 00 , 11 , (+1)0 , 01 .
, , . 3.
12
-
.
1. Bernoulli 1.3. 1.2, .
2. a S = fs 2 Q j s < ag T = ft 2 Q j t > ag. S; T Q s t s 2 S , t 2 T . 2 Q s t s 2 S , t 2 T , a, , . Q .
3. - ( ) - 1 ( ) ( ) 1 . .
4. [] a "1+"
" > 0, a 0.[] a 1 + "+ "2 " > 0, a 1.
5. 0 < a < 1. ax at < < as s; t 2 Q,s < x < t. ax a > 1.
6. n 2 N (1 + a)n 1 + na + n(n1)2
a2 a 0 (1 + a)n 1 + na+ n(n1)
2a2 + n(n1)(n2)
6a3 a 1. .
7. [] n 2 N , xn < yn x < y.[] n 2 N , xn < yn jxj < jyj jyj < x < jyj.
8. . .
[]
xn yn = (x y)(xn1 + xn2y + + xyn2 + yn1) n 2 N, n 2
xn+yn = (x+y)(xn1xn2y+ xyn2+yn1), n 2 N, n 3 .[] x2 + y2 > 0, x2 + xy + y2 > 0 x4 + x3y + x2y2 + xy3 + y4 > 0. ; x3 + x2y + xy2 + y3 > 0 x5 + x4y + x3y2 + x2y3 + xy4 + y5 > 0; ;
[] x; y 0, x 6= y,
n(minfx; yg)n1 xnynxy n(maxfx; yg)n1 ;
maxfx; yg, minfx; yg , , x, y.[] Bernoulli.
9. . O 0! = 1 n! = 1 2 n n 2 N.
nm
= n!
m!(nm)! m;n 2 Z, 0 m n.
13
-
[] nm
n 2 Z, n m
nm
m 2 Z, 0 m n.
[] nm
+
nm1
=n+1m
m;n 2 Z, 1 m n.
n, .
[] Newton:
(x+ y)n =n0
xn +
n1
xn1y + + n
n1xyn1 +
nn
yn =
Pnk=0
nk
xnkyk; n 2 N:
n = 1; 2; 3; 4; 5; 6. Newton , , 10.2 10.3.
: .
[] n 2 N Pnk=0 nk = 2n , Pnk=0 nk(1)k = 0. n = 0;10. , :
[] a 0, n;m 2 N, npa npb = npab np
mpa = m
pnpa = nm
pa.
[] n 2 N, 0 a < b, npa < npb.11. n 2 N , npan = jaj. n ;12. n
pa+ b npa + npb n 2 N, n 2, a; b 0.
a = 0 b = 0.
13. p3 , 7p129 3
p2 +
p5 .
14. 1.4.
[] n; k;m 2 N k, m > 1. nq
km
k;m n- .
[] a0; a1; : : : ; an 2 Z k;m 2 N > 1. km
anxn + an1xn1 + + a0 , k a0
m an .
1.4 .
1.4. a > 0, a 6= 1. y > 0 x ax = y.. a > 1, y 1. U = fu j au yg V = fv j av yg., 0 2 U . , 1.2, n 2 N an > y , ,n 2 V . U; V . u 2 U , v 2 V au y av , au av , a > 1, u v. U; V , u v u 2 U , v 2 V . a = y. n 2 N. 1
n< , 1
n V . a
1n < y , ,
a
y< a
1n . n 2 N, , 1.2, a
y 1. a y.
, < + 1n, + 1
n U . y < a+
1n , , y
a< a
1n .
, 1.2, ya 1. y a .
a y y a a = y.
14
-
. a > 1, 0 < y < 1, , 1
y> 1, a = 1
y, , =
a = a = y. 0 < a < 1, , 1
a> 1, ( 1
a) = y, =
a = a = y., ax = y . a1 = y, a2 = y, a1 = a2 , 1 = 2 .
y 0 a > 0, a 6= 1, x ax = y.. y > 0 a > 0, a 6= 1, ax = y, 1.4, y a loga y.
9 1.5 loga y. , a > 0, a 6= 1, ax loga y . 1.6 .
1.6. a; b > 0, a; b 6= 1.1. loga(yz) = loga y + loga z y; z > 0.2. loga
yz= loga y loga z y; z > 0.
3. loga(yz) = z loga y y > 0 z.
4. logb y =loga yloga b
y > 0.5. loga 1 = 0, loga a = 1.6. 0 < y < z. (i) loga y < loga z, a > 1, (ii) loga y > loga z, 0 < a < 1.
. 1. x = loga y, w = loga z, ax = y, aw = z. ax+w = axaw = yz,
loga(yz) = x+ w = loga y + loga z.2. loga
yz+ loga z = loga(
yzz) = loga y, loga
yz= loga y loga z.
3. x = loga y, ax = y. azx = (ax)z = yz , , loga(y
z) = zx =z loga y.4. x = logb y, w = loga b, b
x = y, aw = b. awx = (aw)x = bx = y. loga y = wx = loga b logb y.5. loga 1 = 0 a
0 = 1 loga a = 1 a1 = a.
6. x = loga y, w = loga z, y = ax , z = aw . ax < aw , a > 1,
x < w , 0 < a < 1, x > w.
6 1.6 loga y : (i) (0;+1), a > 1, (ii) (0;+1), 0 < a < 1.
.
1. 1.6;
2. a > 0, a 6= 1. log 1ay = loga y y > 0.
3. a > 0, a 6= 1. logaz(yz) = loga y y > 0 z 6= 0.4. a1; : : : ; an > 0, a1; : : : ; an 6= 1. loga1 a2 loga2 a3 logan1 an.5. m;n 2 N, m;n 2. , m, n,
logm n .
15
-
1.5 Supremum inmum.
. - A. A u u a a 2 A , , A (1; u]. u A. , A l l a a 2 A , , A [l;+1). l A. , A , l; u A [l; u].
u A, u0 u , , A , l A, l0 l A.
. 1. [a; b] u b . (a; b], [a; b), (a; b), (1; b], (1; b). : , b.
2. [a; b], (a; b], [a; b), (a; b), (a;+1), [a;+1) l a . : , a.
3. , (a;+1), [a;+1), (1;+1) (1; b), (1; b], (1;+1) .4. N , 1 , N l 1 . , 1 N. , 1.1 , , N : u N u n 2 N n > u. , N !!!
1.5. - A.1. A , A .2. A , A .
. 1. U = fu ju Ag. U A. , a u a 2 A, u 2 U . , a u a 2 A, u 2 U . a a 2 A, A. u u 2 U , A.2. : U , L = fl j l Ag.. -, A inmum A infA g.l.b.A. -, A supremum A supA l.u.b.A.
. [a; b], (a; b), (a; b], [a; b) (1; b], (1; b) supremum, b. , [a; b], (a; b), (a; b], [a; b), (a;+1), [a;+1) inmum, a.
16
-
. , , A minimum A minA. , , A maximum A maxA.
1.7. 1. maxA, supA = maxA.2. minA, infA = minA.
. 1. maxA A. A maxA, maxA A.2. .
. 1. A = f0g [ [2; 3] [ f4g minA = 0 maxA = 4. infA = 0 supA = 4.
2. minN = 1, infN = 1.3. A = f 1
njn 2 Ng = f1; 1
2; 13; 14; : : : g maxA = 1, supA = 1. A
. infA., l 0 A. l > 0, , , n 2 N 1
n< l, l A.
A - , A 0. , infA = 0.
. - A infA = 1. - A , supA = +1.
. A , . , +1 , , , , , A.
. N , supN = +1.. , supA, , +1. , , supA = u ( ) supA u, supA = u 2 R . infA.
1.8 , , supremum inmum.
1.8. - A.1. a supA a 2 A. , u < supA a 2 A a > u ,, u < a supA.2. a infA a 2 A. , l > infA a 2 A a < l ,, infA a < l.. 1. A , supA ( ) A, a supA a 2 A. A , supA = +1 , , a supA a 2 A., u < supA. supA A, u A, a 2 A a > u.2. .
1.9. 1. - A supremum inmum infA supA.2. - A, B A B, infB infA supA supB.
17
-
. 1. - A , , 1.5, supA . A , , , supA = +1. , - A , infA , , A , infA = 1., a 2 A. infA a supA, infA supA.2. B , supB = +1, supA supB . B , supB < +1. a 2 A a 2 B, a supB. supB A , , supA supB. infB infA .
. A , infA, supA , , infA = supA. A , infA < supA.
1.5 .
supremum. -, .
, supremum . -, 1.10 supremum. supremum .
1.10. -, . .
. A;B - a b a 2 A, b 2 B., b 2 B A , B , A . A , supA. supA A, a supA a 2 A. b 2 B A supA A, supA b b 2 B., = supA, a b a 2 A, b 2 B.
, , inmum, -, . 1.5 . 1.10, inmum . 14.
. A . A :
x1; x2 2 A, x1 < x2 x x1 < x < x2 x 2 A. : . 1.11 , - R, .
1.11. - A : x1; x2 2 A, x1 < x2 x x1 < x < x2 x 2 A. A .. u = supA l = infA, 1 l u +1. , , A [l; u]. x 2 (l; u). x A, x1; x2 2 A, x1 < x < x2 . , x 2 A. , (l; u) A. (l; u) A [l; u] : A = (l; u),A = [l; u], A = (l; u] A = [l; u). A .
18
-
.
1. 1.5 1.7, 1.8 1.9.
2. maxfx; yg = x+y+jxyj2
minfx; yg = x+yjxyj2
.
3. [] t x, t y t z t minfx; y; zg.[] t x, t y t z t maxfx; y; zg.[] t x, y, z;
4. (a;+1), (a; b), (a; b] l a; (1; b), (a; b), [a; b) u b;
5. [] inma suprema f1; 0; 2; 5g, [1; 5], (1; 5), (1; 0] [ (2; 5]. ;
[] infA = infB supA = supB. A = B;
6. inma suprema f(1)nn jn 2 Ng, f1+(1)n2n
+ 1(1)n
2n jn 2 Ng,
f(1)n + 1njn 2 Ng, fn(1)n(n1)
2njn 2 Ng, f 1
n+ 1
mjn;m 2 Ng, f 1
n 1
mjn;m 2 Ng,
fx + y j 0 < y < 1; 4 < x < 5g, fx y j 0 < y < 1; 4 < x < 5g, S+1n=1[2n 1; 2n],S+1n=1[
12n; 12n1 ], fx j sin x > 0g, fx > 0 j sin 1x > 0g.
7. - A. , supA, A, : supA = +1, supA < +1. A infA.
8. [] a < b A = fr 2 Q j a < r < bg. , infA supA. A = fx 2 R nQ j a < x < bg ;[] A = f r
r2+2j r 2 Qg. infA supA.
9. [] a > 1, supfas j s 2 Q; s < xg = ax = inffat j t 2 Q; t > xg. - - ax x. a = 1 0 < a < 1;
: 1.3.
[] y > 0, a > 1, supfu j au yg = loga y = inffv j av yg. - - loga y. 0 < a < 1;
10. A = [0; 2], A = [0; 2), A = [0; 1] [ f2g 1.8, , , . - A u = supA.
[] A \ (u "; u] 6= ; " > 0;[] A \ (u "; u) 6= ; " > 0;[] [], [] , , u /2 A;[] l = infA.
11. - A.
[] supA u a u a 2 A.[] u supA < u a 2 A, a > .[] infA.
19
-
12. - A;B.
[] supA infB a b a 2 A, b 2 B.[] a b a 2 A, b 2 B. , supA, infB, : a b a 2 A, b 2 B.[] a b a 2 A, b 2 B " > 0 a 2 A, b 2 B b a ". supA = infB. 5 1.2.
13. - A;B. A supA supB a 2 A < a b 2 B, b > .
14. [] inmum.
[] , supremum inmum .
15. supremum inmum .
[] A, B -, sup(A [ B) = maxfsupA; supBg inf(A [ B) = minfinfA; infBg. , sup(A \ B) minfsupA; supBg inf(A \ B) maxfinfA; infBg.[] . - A A = fa j a 2 Ag. sup(A) = infA inf(A) = supA.[] . - A;B A+B = fa+ b j a 2 A; b 2 Bg. sup(A+B) = supA+ supB inf(A+B) = infA+ infB.
: a+b supA+supB a 2 A, b 2 B. sup(A+B) supA+supB. a+b sup(A+B) a 2 A, b 2 B. , a 2 A, b sup(A+B)a b 2 B. , a 2 A, supB sup(A+ B) a. , a 2 A, a+ supB sup(A+B). , : supB = +1, supB 2 R.[] . - A;B A B = fab j a 2 A; b 2 Bg. A;B (0;+1), inf(A B) = infA infB sup(A B) = supA supB. A;B [0;+1) A, B 0;
16. ;; , inf ;, sup ;; inf ; sup ;;
17. 11 2.5.
[] f : [a; b]! R [a; b]. f(a) > a f(b) < b, 2 (a; b) f() = .: A = fx 2 [a; b] jx f(x)g. A - = supA.
[] I f : I ! R x 2 I 0 > 0 f(x0) f(x) f(x00) x0; x00 2 (x 0; x+ 0) \ I , x0 < x < x00 . f I .
: a; b 2 I , a < b. A = fx 2 [a; b] j f(a) f(x)g. A - . = supA , , 2 A = b. f(a) f(b).
20
-
2
.
2.1 .
2.1.1 , , .
. ( ) x : N! R N . n 2 N x(n) , , xn . , xn = x(n) n 2 N.
, : x1 , x2 , x3 . , / . xn+1 xn xn1 xn . n, - N, . x : N! R (x1; x2; : : : ; xn; : : : ), (xn), (xn)+1n=1 ., , , x, n, : (yn), (xk), (zm) .
. 1. ( 1n) (1; 1
2; 13; : : : ; 1
n; : : : ).
2. (n) (1; 2; 3; 4; : : : ; n; : : : ).
3. (1) (1; 1; 1; : : : ; 1; : : : ).
4. ((1)n1) (1;1; 1;1; : : : ; 1;1; : : : ).5.
1
10n
( 1
10; 1102
; 1103
; : : : ; 110n
; : : : ).
6. n- n, (1; 2; 2; 3; 2; 4; 2; 4; 3; 4; 2; : : : ).
7. (m n)+1n=1 (m 1;m 2;m 3; : : : ;m n; : : : ).8. (m n)+1m=1 (1 n; 2 n; 3 n; : : : ;m n; : : : ). (xn)
+1n=1 , (xn), -
- .
: . , , . (1)+1n=1 f1g . , , (1; 1; 1; : : : ). , . , . , (1;1; 1;1; 1;1; : : : ),
21
-
(1; 1;1; 1; 1;1; 1; 1;1; : : : ) f1; 1g. : (1; 1
2; 13; 14; 15; 16; : : : ), (1
2; 1; 1
4; 13; 16; 15; : : : )
f1; 12; 13; 14; 15; 16; : : : g = f 1
njn 2 Ng.
. (xn) xn+1 xn n 2 N, xn+1 > xn n 2 N, xn+1 xn n 2 N, xn+1 < xn n 2 N. (xn) . (xn) , c xn = c n 2 N.
, . , , . .
. (xn) , u xn u n 2 N. u (xn). (xn) , l xn l n 2 N. l (xn). (xn) , l; u l xn u n 2 N.
, u (xn), u0 u
, l (xn), l0 l .. 1. (c) .2. ( 1
n), (1)n1
n
, (n1
n), ((1)n1)
[1; 1].3.
(1+(1)n1)n2
(1; 0; 3; 0; 5; 0; 7; 0; : : : ) -
. , l 0 . - - u . (1+(1)
n1)n2
u n 2 N,, n = 2k 1, 2k 1 u k 2 N. k u+1
2
k 2 N. .4. (1; 0;3; 0;5; 0;7; 0; : : : ), , . l , l, l, .
5. ((1)n1n) (1;2; 3;4; 5;6; : : : ) -. , u l , , (1)n1n u n 2 N l (1)n1n n 2 N. , n 2 N,.
(xn) , l; u l xn u n 2 N. M = maxfu;lg, M u M l , ,M xn M , , jxnj M n 2 N. , (xn) , M jxnj M n 2 N. : M jxnj M n 2 N, M xn M n 2 N, M M (xn). : (xn) M jxnj M n 2 N.
22
-
2.1.2 n 2 N . n 2 N.
n 2 N. : n 234 4 n n2 n > 8 xn < xn+1 (xn).
. , , n 2 N n0 2 N n 2 N, n n0 .
n (xn) (xn), (xn) , , n 2 N . : xn+1 xn, xn+1 > xn,xn u, xn = c. , (xn) , , , , u, c.
. 1. (1; 23;p2 ;2;1;1;1;1; : : : ) ,
.
2. (n214n+8) . , n214n+8 = (n7)241 n 2 N n 7 , , .
n 2 N. n0 2 N n 2 N, n n0 n00 2 N, n00 n0 , n 2 N, n n00 .
, n 2 N. n00; n000 2N n 2 N, n n00 n 2 N, n n000 . n0 = maxfn00; n000g 2 N. n0 n00 , n 2 N, n n0 . , n0 n000 , n 2 N, n n0 . , , n 2 N, n n0 . :
, , , .
. n2 3n 37 n 2 N, n p157+32
, , n 2 N,n 8. , 2n+1
n+1> 25
13 n 2 N, n > 12 , , n 2 N, n 13.
n2 3n 37 2n+1n+1
> 2513
n 2 N, n maxf8; 13g = 13. , , :
, n00 ; n000 ,
, . , , !
. n 2 N .. (1)n1 > 0 n 2 N, n 2 N., (1)n1 0 n 2 N, n 2 N. n 2 N n 2 N .
.1. : 1; 4; 9; 16, 25; 36. : 49;
24; ;
: .
23
-
2. [] a+b2
+ (1)n1 ab2
.
[] m 2 N, nm[ nm]+1n=1
.
: m = 1; 2; 3.
3. [] , .
[] .
4. [] (xn) (yn) (xn + yn). , , . , .
[] (xn) (yn) (xnyn). (i) , , , xn; yn 0 n 2 N.
5. ((1)n1n), (1)n1n
,
18nn2+n+1
,13n
n!
,n30
2n
,2[n
2],n 3[n
3]
; ; ;
6. , , , , . , () .
7. (xn) . - x xn = x n 2 N. . -: 11 2.2, 2 12 2.7 2 2.9.
8. . a; b; p; q, p; q 0. (xn) x1 = a x2 = b xn+2 = pxn+1 + qxn n 2 N. n- xn .
[] p 6= 0; q = 0. xn = bpn2 n 2 N, n 2.[] p = 0; q 6= 0. xn = aq n12 n 2 N xn = bq n22 n 2 N.[] p 6= 0; q 6= 0. x2 = px+ q. = p2 + 4q > 0, , 1 =
p+p
2, 2 =
pp2
. ; + = a, 1 + 2 = b. xn = 1
n1 + 2n1 n 2 N. = p2 + 4q = 0, , = p
2. ; = a, + = b.
xn = n1 + (n 1)n1 n 2 N.
= p2+4q < 0 ( q < 0), , 1 =p+ip2
,
2 =pip
2. =
pq > 0. p2
2+p
2
2= 1, 2 [0; 2)
cos = p2
, sin =p2
. 1 = (cos + i sin ), 2 = (cos i sin ). 2 cos 2 = p cos + q, 2 sin 2 = p sin . ; = a, ( cos + sin ) = b xn =
n1 cos(n 1) + n1 sin(n 1) n 2 N.
24
-
[] n- x1 = x2 = 1 xn+2 = 3xn , xn+2 = xn+1+xn ,xn+2 = 2xn+1xn , xn+2 = xn+1xn . Fibonnaci 1; 1; 2; 3; 5; 8; 13.
9. - - (xn) xn+1 = x1 + + xn ; xn+3 =
xnxn+2xn+1
, xn+1 = 1 1xn xn+1 = 2 1xn .10. :
[] n 2 N k 2 N n 2 N, n > k .[] n 2 N n 2 N.[] n 2 N n 2 N .
2.2 .
. (xn) x (xn) x x (xn) " > 0 n0 2 N jxn xj < " n 2 N, n n0 . : (xn) x " > 0 jxn xj < ". (xn) x xn ! x limxn = x limn!+1 xn = x. (xn) , (xn) .
: (xn) x n- xn x . : (xn) x n- xn x n .
. xn ! x, " > 0 n0 2 N - " - n 2 N,n n0 ()) jxn xj < " , , jxn xj < " (() n 2 N, n n0. :
(n 2 N; n n0) ) (jxn xj < ")
(jxn xj < ") ( (n 2 N; n n0):. 1. ( 1
n) 0. 1
n! 0.
, " > 0. n0 2 N j 1n 0j < " n 2 N, n n0 . n 2 N, j 1n 0j < " 1n < " n > 1" . : (j 1
n 0j < ")( ( 1
n< ")( (n > 1
").
1.1 n0 2 N, n0 > 1" . , n > 1" n0 n 2 N, n n0 . n0 2 N n 2 N, n n0 n > 1
", , j 1
n 0j < ". : (n 2 N; n n0) ) (n > 1") ) ( 1n < ") )
(j 1n 0j < ").
- - n0 2 N.
25
-
2.1. a 0, n0 = [a]+1 n 2 N, n > a. a < 0, n0 = 1 n 2 N, n > a.. .
: n 2 N, n > 3 1, n 2 N, n > 83 3 = [8
3]+1
n 2 N, n > 2 3 = 2 + 1 = [2] + 1. . n0 = [
1"] + 1 n 2 N, n > 1
". ,
n0 2 N, j 1n 0j < " n 2 N, n n0 .2. (c) c.
c! c:
" > 0. n0 2 N jc cj < " n 2 N,n n0 . jc cj < " 0 < ". , 0 < " n 2 N. n0 = 1 2 N jc cj < " n 2 N, n n0 .3. ((1)n1) , . - - ((1)n1) x. " > 0 n0 2 N j(1)n1 xj < " n 2 N, n n0 . , n0 , n 2 N, n n0 . n 2 N, n n0 j 1 xj < " n 2 N, n n0 j1 xj < ". j1xj < " 1 2 (x "; x+ ") j1xj < " 1 2 (x "; x+ ")., " > 0, 1, 1 2"., , 0 < " 1, 1, 1 2. , .
. (xn) +1 (xn) +1 +1 (xn) M > 0 n0 2 N xn > M n 2 N,n n0 . : (xn) +1 M > 0 xn > M . (xn) +1, xn ! +1 limxn = +1 limn!+1 xn = +1 ., (xn) 1 (xn) 1 1 (xn) M > 0 n0 2 N xn < M n 2 N,n n0 . : (xn) 1 M > 0 xn < M . (xn) 1, xn ! 1 lim xn = 1 limn!+1 xn = 1 .
: (xn) +1 n- xn . : (xn) +1 n- xn +1 n . 1.
7, 8 9 .
. 1. (n) +1. n! +1. M > 0. n0 2 N n > M n 2 N,n n0 . 1.1, n0 2 N, n0 > M . , n > M n0 n 2 N, n n0 . - - n0 = [M ] + 1 2 N , n0 , n > M n 2 N, n n0 .2. (
pn) +1. pn! +1.
M > 0. n0 2 N pn > M n 2 N, n n0.
26
-
M > 0, pn > M n > M2. : (
pn > M)( (n > M2).
n0 2 N n0 > M2 , , n 2 N, n n0 n > M2, ,
pn > M . : (n 2 N; n n0)) (n > M2)) (
pn > M).
3. , n < M pn < M , : n! 1 pn! 1.4. ((1)n1n) (1;2; 3;4; 5;6; : : : ) +1 1. - - +1. M > 0 n0 2 N (1)n1n > M n 2 N, n n0 . n0 , n 2 N, n n0 . n 2 N, n n0 n > M n 2 N, n n0 n > M . , n > M ,, , . , 1 .
, , .
. 1. ( 1n).
na ! 0; a > 0:
" > 0. n0 2 N j 1na 0j < " n 2 N, n n0 . n 2 N, j 1
na 0j < " 1
na< " na > 1
" n > (1
")1a .
n0 2 N n0 > (1")1a , , n 2 N, n n0 n > (1")
1a
, , j 1na 0j < ".
2. (n) (pn).
na ! +1; a > 0:
M > 0. n 2 N, na > M n > M 1a . n0 2 N n0 > M
1a , n 2 N, n n0 n > M 1a , , na > M .
3. loga n! +1; a > 1:
M > 0. n 2 N, loga n > M n > aM . n0 2 N n0 > aM , n 2 N, n n0 n > aM , , loga n > M .4. . (a; a2 ; a3 ; a4 ; : : : ), (an). - : a.
(i) a = 1, (1) 1. , a = 0, (0) 0.
(ii) a 1, (an) a 1, a2 1, a3 1, a4 1; : : : . (an) . , (an) x. " > 0 n0 2 N jan xj < " , , an 2 (x "; x + ") n 2 N, n n0 . , an n 2 N, n n0 an n 2 N, n n0 2. , 0 < " 1, . , (an) +1. M > 0 n0 2 N an > M n 2 N, n n0 . n 2 N, n n0 ., (an) 1.(iii) 0 < jaj < 1, (an) 0. " > 0. , jan 0j < " jajn < " n > logjaj ". n0 2 N
27
-
n0 > logjaj " , , n 2 N, n n0 n > logjaj " , , jan 0j < ".(iv) a > 1, +1. M > 0. , an > M n > logaM . n0 2 N n0 > logaM , , n 2 N, n n0 n > logaM , , an > M .
an
8>>>>>:! +1; a > 1! 1; a = 1! 0; 1 < a < 1 ; a 1
2.4 6 2.5 .
. . - 2.3 2.23 - , .
. ! , lim, limn!+1 - , 1. , 1. limn!+1 xn , , R . , , limn!+1 xn = x ( ), limn!+1 xn x, limn!+1 xn = x 2 R .
.
1. , , : limn!+1 n2pn,
limn!+1(1)n8n
32n, limn!+1 log3 n, limn!+1
(1)n22n3n
.
2. x limn!+1(x+1)2n
(2x+1)n;
3. , : 1n+8
! 0, 3n+12n+5
! 32, 1p
n+5! 0,
n2 7n! +1, 2n + 2n ! +1, 3+log2 n1+3 log2 n
! 13.
4. 0 +1, , , . :
[] 3+(1)n
2n! 0, 3+(1)n
2n> 0 n 2 N
3+(1)n2n
.
[] (3(1)n1)n
2! +1 (3(1)n1)n
2
.
5. limm!+1limn!+1(cosm!x)2n
=
(1; x 2 Q0; x 2 R nQ
: x 2 Q m 2 N , m!x .
28
-
6. [] xn ! x. " > 0 n0(") n0 2 N jxnxj < " n 2 N, n n0 . , 0 < "0 < ", n0("0) n0(").[] xn ! +1. M > 0 n0(M) n0 2 N xn > M n 2 N, n n0 . , M 0 > M > 0, n0(M 0) n0(M).
7. :
[] (xn) x "0 > 0 jxn xj "0 n 2 N.[] (xn) +1 M0 > 0 xn M0 n 2 N.[] (xn) 1 M0 > 0 xn M0 n 2 N.
8. .
[] "0 > 0, xn ! x ", 0 < " "0 jxnxj < ". : , - " > 0 "0 > 0, .
[] M0 > 0, xn ! +1 M M0 xn > M . .
[] xn ! +1 M - - xn > M .
9. .
[] , " > 0 jxn xj < ", " jxn xj ". . : xn = (1)n1 , x = 0, " = 1. : xn ! x " > 0 jxn xj ".[] , M > 0 xn > M , M xn M . . : xn = 1, M = 1. : xn ! +1 M > 0 xn M .
10. : .
(xn) x : n0 2 N " > 0 jxn xj < " n 2 N, n n0 . ; xn ! x.
11. [] (xn) . , xn ! x, (xn) x .: A (xn) d > 0 A. n0 2 N jxn xj < d2 ,, xn 2 (x d2 ; x+ d2) n 2 N, n n0 . (x d2 ; x+ d2) A.
[] (xn) xn 2 Z n 2 N. xn ! x, (xn) x 2 Z.
29
-
2.3 .
, - .
. " > 0. (x "; x + ") "- x Nx(") = (x "; x+ ").
", x Nx . , Nx x : : : " > 0 Nx(") x : : : Nx x : : : " > 0, Nx(") x : : : .
. " > 0. (1";+1] "- +1 [1;1
")
"- 1. N+1(") = (1" ;+1] N1(") = [1;1"). ", 1
N1 . M = 1" , " =1M
(M;+1] (1
";+1] [1;M)
[1;1").
, " x 2 R , Nx(") . 1. " x Nx("). " x Nx(").
jxn xj < " xn ! x , ,x " < xn < x+ " , , xn 2 (x "; x+ ") , , xn 2 Nx(")., xn > M xn < M xn ! 1 ,, xn 2 (M;+1] xn 2 [1;M) , , xn 2 N+1(") xn 2 N1("), " = 1
M.
, , , .
. x 2 R . xn ! x " > 0 n0 2 N xn 2 Nx(") n 2 N, n n0 , , " > 0 xn 2 Nx("). : xn ! x Nx x xn 2 Nx .
. , , 1, . , .
.1. x 2 R .
[] , 0 < "1 "2 , Nx("1) Nx("2).[] > 0 n 2 N Nx( 1n) Nx(").
2. x 2 R .[]
T">0Nx(") = fxg , ,
x x.
[] T+1
n=1Nx(1n) = fxg.
3. x; y 2 R , x 6= y. " > 0 Nx(") \Ny(") = ;.
30
-
2.4 .
2.1 , , . 6 2.1 .
2.1. (xn), (yn) . , .
. (xn), (yn) k0;m0 2 N xk0 = ym0 , xk0+1 = ym0+1 , xk0+2 = ym0+2 ; : : : : xn ! a 2 R yn ! a. " > 0. n0 2 N xn 2 Na(") n 2 N, n n0 . n0 k0 , xn 2 Na(") n 2 N, n k0 . yn 2 Na(") n 2 N, n m0 . n0 > k0 , p = n0 k0 , n0 = k0+ p. , xn 2 Na(") n 2 N, n k0 + p. yn 2 Na(") n 2 N, n m0 + p. n0
0 2 N (n00 = m0 n00 = m0 + p) yn 2 Na(") n 2 N, n n00 . yn ! a.. 1. (1; 1
2; 13; 14; 15; 16; : : : ), (2; 5; 1
4; 15; 16; : : : ). -
0. , , , , 0.
2. (xn) . (xn) (x1; x2; x3; : : : ). (x2; x3; x4; : : : ), (xn+1), (xn). (x3; x4; x5; : : : ), (xn+2). , m 2 N, xn ! x 2 R xn+m ! x 2 R . : 1
n+3! 0 log2(n+ 8)! +1.
. , 2.2 2.3 .
2.2. xn ! x 2 R .1. x > u, xn > u.2. x < l, xn < l.3. u < x < l, u < xn < l.
. 1. xn ! x x > u. x u > 0, jxn xj < x u ,, xn > x (x u) = u. xn ! +1. M > 0 M u. xn > M , , xn > u.2. .3. xn > u xn < l. xn > u xn < l.
2.3 - : , . , . 16 2.7 13 2.9.
31
-
2.3. 1. xn l n 2 N xn ! x 2 R, x l.2. xn u n 2 N xn ! x 2 R, x u.3. u < l xn u n 2 N xn l n 2 N, (xn) .
. 1. x < l, xn < l, xn l n 2 N.2. .3. (xn) , u l, l u.. 1. xn ! x 2 R xn 2 [l; u] n 2 N, x 2 [l; u].2. a 1, (an) , an 1 n 2 N an 1 n 2 N.3. ((1)n1n) , (1)n1n 1 n 2 N (1)n1n 1 n 2 N.4.
n 3[n
3] . , n = 3k k 2 N,
n 3[n3] = 0 0 , n = 3k + 1 k 2 N, n 3[n
3] = 1 1.
2.4 . .
2.4. .
. (xn) . a . 2.2, xn > a , , xn < a. xn > a xn < a. .
2.5. xn yn n 2 N xn ! x 2 R, yn ! y 2 R, x y.. x > y. a x > a > y. xn > a a > yn . , xn > a a > yn . xn > yn , , xn yn n 2 N. .. 1
n< 1
n n 2 N 1
n! 0, 1
n! 0.
, xn < yn n 2 N xn ! x 2 R ,yn ! y 2 R , x < y. xn < yn n 2 N xn yn n 2 N, x y.
2.6 2.7 : .
2.6. xn yn .1. xn ! +1, yn ! +1.2. yn ! 1, xn ! 1.. 1. M > 0. xn > M , yn xn , xn > M yn xn . yn > M , ,yn ! +1.2. .
32
-
. 1. n+(1)n1 n1 n 2 N. n1! +1, n+ (1)n1 ! +1.2. n
2+2n+1n+2
n n 2 N n! +1, n2+2n+1n+2
! +1.3. [
pn] >
pn 1 n 2 N pn 1! +1. [pn]! +1.
2.7 .
2.7. xn yn zn . xn ! a zn ! a, yn ! a.. " > 0. jxn aj < " , , jzn aj < ". jxn aj < " jzn aj < " xn yn zn . xn > a " zn < a + " xn yn zn . a " < yn < a + " , , jyn aj < ". yn ! a.. 1. 1
n (1)n1
n 1
n n 2 N. 1
n! 0 1
n! 0,
(1)n1n
! 0.2. , 1
n sinn
n 1
n n 2 N, sinn
n! 0.
2.8. , .
. xn ! x. n0 2 N jxn xj < 1 n 2 N, n n0 . jxn xj < 1 jxnj = j(xn x) + xj jxn xj + jxj < 1 + jxj. jxnj < 1 + jxj n 2 N, n n0 . M = maxfjx1j; : : : ; jxn01j; 1 + jxjg jxnj M n 2 N,n n0 1 jxnj < 1 + jxj M n 2 N, n n0 . jxnj M n 2 N, (xn) .. 2.8. H ((1)n1) - .
2.9. 1. +1, .2. 1, .. 1. xn ! +1. n0 2 N xn > 1 n 2 N,n n0 . l = minfx1; : : : ; xn01; 1g, xn l n 2 N, n n01 xn > 1 l n 2 N, n n0 . xn l n 2 N, (xn) ., M > 0 xn > M . M > 0 (xn), (xn) .2. .
. 1, 2 2.9 . -
(1+(1)n1)n2
, (1; 0; 3; 0; 5; 0; 7; : : : ), .
, +1 (1+(1)n1)n2
0 n 2 N., (1; 0;3; 0;5; 0;7; : : : ) , 1.
(xn) (xn). 2.10. xn ! x 2 R , xn ! x.
33
-
. xn ! x. " > 0. jxn xj < ". j(xn) (x)j = jxn xj < ". xn ! x. xn ! +1. M > 0. xn > M , xn < M ,, xn ! 1 = (+1)., , xn ! 1, xn ! +1 = (1).
(xn) (yn) (xn + yn).
2.11. xn ! x 2 R yn ! y 2 R x+ y , xn + yn ! x+ y.. xn ! x, yn ! y. " > 0. jxn xj < "2 jyn yj < "2 . jxn xj < "2 jyn yj < "2 . ,
j(xn + yn) (x+ y)j = j(xn x) + (yn y)j jxn xj+ jyn yj < "2 + "2 = ", , xn + yn ! x+ y. xn ! +1 yn ! y 2 R [ f+1g. (yn) , l yn l n 2 N. M > 0. M 0 > 0 M 0 M l. xn > M
0 , xn > M l, xn+yn > (M l)+ l =M . xn + yn ! +1 = (+1) + y. .
. 1. 1n! 0 (1)n
n! 0, 1
n+ (1)
n
n! 0 + 0 = 0.
2. n! 1 1n! 0, n2+1
n= n+ 1
n! (1) + 0 = 1.
3. n! +1 pn! +1, n+pn! (+1) + (+1) = +1. (+1)+ (1),
+1 1 R :
. 1. n+ c! +1, n! 1 (n+ c) + (n) = c! c.2. 2n! +1, n! 1 2n+ (n) = n! +1.3. n! +1, 2n! 1 n+ (2n) = n! 1.4. n+ (1)n1 ! +1, n! 1 (n+ (1)n1) + (n) = (1)n1 .
(xn) (yn) (xn yn). (xn yn) , xn yn = xn + (yn). 2.12. xn ! x 2 R yn ! y 2 R x y , xn yn ! x y.
(xn) (yn) (xnyn).
2.13. xn ! x 2 R yn ! y 2 R xy , xnyn ! xy.. xn ! x, yn ! y. " > 0. jxn xj < "3jyj+1 , , jyn yj < minf "3jxj+1 ; 13g. jxn xj < "3jyj+1 jyn yj 0. xn >
Ml
, , yn > l. xn >Ml
yn > l. xnyn >
Mll =M , , xnyn ! +1 = (+1)y.
.
. 1. 1n! 0 (1)n
n! 0, (1)n
n2= 1
n(1)nn
! 0 0 = 0.2. n1
n! 1 1
n! 0, n1
n2= n1
n1n! 1 0 = 0.
3. n! +1 1n! 1, nn2 = n(1n)! (+1)(1) = 1. (n n2) .4. c xn ! x 2 R cx , , , c = 0 x = 1. , c! c, cxn ! cx., c = 0, , x 2 R , cxn = 0xn = 0! 0.5. a > 0. c > 0, cna ! c(+1) = +1. c < 0, cna ! c(+1) = 1.6. a > 0, cna ! c0 = 0.7. n. a0+a1x+ +akxk , ak 6= 0, k 1. a0 + a1n+ + aknk = aknk
a0ak
1nk
+ a1ak
1nk1 + + ak1ak 1n + 1
.
1, 0. ,akn
k ! ak(+1).
a0 + a1n+ + aknk ! ak(+1)1 =(+1; ak > 01; ak < 0
., limn!+1(a0 + a1n+ + aknk) = limn!+1 aknk . : 3n2 5n+ 2! +1 1
2n5 + 4n4 n3 ! 1.
8. . a (1 + a+ a2 + + an1 + an), (1 + a; 1+ a+ a2 ; 1+ a+ a2 + a3 ; : : : ). :
1 + a+ a2 + + an8>:! +1; a 1! 1
1a ; 1 < a < 1 ; a 1
.
a > 1, 1 + a+ a2 + + an = an+11a1 ! (+1)1a1 = +1.
a = 1, 1 + a+ a2 + + an = n+ 1! +1. 1 < a < 1, 1 + a+ a2 + + an = an+11
a1 ! 01a1 = 11a . a 1. : an+1 = 1 + (a 1)(1 + a + a2 + + an) n 2 N. 1 + a +a2 + + an ! x 2 R, an+1 ! 1 + (a 1)x. , (an+1) . : 1 + a + a2 + + an = an+11
a1 21a n 2 N 1 + a + a2 + + an = an+11
a1 0 n 2 N. 0 < 21a , .
2.14. 1. xn ! x 2 R k 2 N, xnk ! xk .2. k 2 N, k 2. xn 0 n 2 N xn ! x 2 R , kpxn ! k
px .
35
-
. 1. xnk = xn xn ! x x = xk .
2. x 0. x = 0. " > 0. jxn 0j < "k , xn 0 n 2 N, 0 xn < "k . j kpxn 0j = kpxn < ". kpxn ! 0. x = +1. M > 0. xn > Mk . kpxn > M , kpxn ! +1.
0 < x < +1. " > 0. "0 = minf"; kpxg > 0. ( kpx "0)k < x 0. jxn xj < lx". xn > l jxn xj < lx". , 1
xn 1
x
= jxnxjxnx
< lx"lx
= ";
1xn! 1
x.
xn ! +1. " > 0. xn > 1" , 0 < 1xn < " ,,
1xn 0 = 1
xn< ". 1
xn! 0 = 1
+1 ., 1.. 1. a > 1, 1
loga n! 1
+1 = 0.
2. xn ! 0. , (1)n1n ! 0, n(1)n1 = (1)n1n . 1
0 .
. , ; 2.16.
2.16. xn 6= 0 n 2 N xn ! 0.1. xn > 0,
1xn! +1.
2. xn < 0, 1xn! 1.
36
-
. 1. M > 0. jxn 0j < 1M . xn > 0, jxn 0j < 1M xn > 0. 0 < xn < 1M . 1
xn> M , 1
xn! +1.
2. .
(xn) (yn) xnyn
. (xn
yn)
, xn
yn= xn
1yn
.
2.17. yn 6= 0 n 2 N. xn ! x 2 R yn ! y 2 R xy , xn
yn! x
y.
. 1. n. a0+a1x++akxkb0+b1x++bmxm , ak 6= 0,
bm 6= 0. a0+a1n++akn
k
b0+b1n++bmnm =akn
k
bmnm
a0ak
1
nk+a1ak
1
nk1++ak1ak
1n+1
b0bm
1nm
+b1bm
1nm1++
bm1bm
1n+1
.
1. akn
k
bmnm= ak
bmnkm ,
a0+a1n++aknkb0+b1n++bmnm !
8>:akbm(+1); k > m
akbm; k = m
0; k < m
. : n
32n2+n+12n23n1 ! +1, n
2+nn+2
! 1, n4n3n4+1
! 1 n2+n+4n3+n2+5n+6
! 0.2. 2n
3+n2+n+12n+3
! 1, 2n3+n2+n+12n+3
7 ! (1)7 = 1.3. n
3+n+73n3+n2+1 ! 13 ,
n3+n+73n3+n2+1
3 ! (13)3 = 1
27.
, 00 +1
+1 , 0 +1 R, , [0;+1], , .
. 1. cn! 0, 1
n! 0 ( c
n)/( 1
n) = c! c.
2. 1n! 0, 1
n2! 0 ( 1
n)/( 1
n2) = n! +1.
3. 1n2! 0, 1
n! 0 ( 1
n2)/( 1
n) = 1
n! 0.
4. 1n! 0, 1
n2! 0 ( 1
n)/( 1
n2) = n! 1.
5. 1n 2+(1)n1
n 3
n n 2 N, 2+(1)n1
n! 0. , 1
n! 0
(2+(1)n1
n)/( 1
n) = 2 + (1)n1 , 2 + (1)n1 1 n 2 N
2 + (1)n1 3 n 2 N.6. 1 c > 0, 2, 3 5 (xn) (yn) +1 [0;+1] .
(xn) (jxnj). 2.18. xn ! x 2 R , jxnj ! jxj.
37
-
. xn ! x. " > 0. jxn xj < ". jjxnj jxjj jxn xj < " , , jxnj ! jxj. xn ! +1 xn ! 1. M > 0. xn > M xn < M ,. , jxnj > M , jxnj ! +1 = j 1j.
. 2.18 . ,(1)n1 = 1! 1
(1)n1 . , , 12[]. - - .
6 2.5. , , 18[]. , , 5.8.2 l Hopitl.
. 1. a > 0.
npa! 1; a > 0:
a = 1 : np1 = 1! 1.
a > 1. : " > 0. j npa1j < " ( npa > 1) npa1 < " n
pa < 1+" 1
n< loga(1+") n >
1loga(1+")
.
n0 2 N n0 > 1loga(1+") . n 2 N, n n0 n >1
loga(1+"),
, j npa 1j < ". npa! 1. : Bernoulli (1 + a1
n)n 1 + na1
n= a ,
, 1 npa 1 + a1n
n 2 N. npa! 1. 0 < a < 1, 1
a> 1, n
pa = 1
np
1/a! 1
1= 1.
2. npn! 1:
Bernoulli, (1+pn1n
)n 1+npn1n
=pn , 1 npn (1+
pn1n
)2 1.
an
n! +1; a > 1:
: Bernoulli, (pa)n = (1 +
pa 1)n 1 + n(pa 1) >
n(pa 1) , , an
n> n(
pa 1)2 n 2 N. an
n! +1.
: b 1 < b < a. Bernoulli bn = (1+ b 1)n 1 + n(b 1) > n(b 1) n 2 N. an
n= b
n
n(ab)n > (b 1)(a
b)n
n 2 N , ab> 1, a
n
n! +1.
4. an
n!! 0:
n0 = [jaj]+1 2 N. n 2 N, n n0 jann! j = jajn01
(n01)!jajnn0+1n0n
jajn01(n01)!(
jajn0)nn0+1 .
0 jajn0< 1, ( jaj
n0)nn0+1 ! 0. jan
n!j ! 0 , jan
n!j an
n! jan
n!j
n 2 N, ann!! 0.
5. (an) - a > 1, jaj < 1.
38
-
a > 1. : b = a 1 > 0. Bernoulli, an = (1 + b)n 1 + nb > nb n 2 N. nb! +1, an ! +1. : an > nb. M > 0. , an > M nb > M n > M
b. n0 2 N
n0 >Mb. n 2 N, n n0 n > Mb , , an > M .
an ! +1. : jaj < 1, : 0 < 1
a< 1, 1
an= ( 1
a)n ! 0 , , an ! +1.
jaj < 1. a = 0 , 0 < jaj < 1. : 1jaj > 1, , ,
1jajn = (
1jaj)
n ! +1. jajn ! 0. , jajn an jajn n 2 N, an ! 0. : b = 1jaj 1 > 0. Bernoulli, jajn = 1(1+b)n
11+nb
< 1nb
n 2 N. 1nb< an < 1
nb n 2 N, an ! 0.
: " > 0. jajn < 1nb
jan 0j < " 1
nb< " n > 1
"b. n0 2 N n0 > 1"b .
n 2 N, n n0 n > 1"b , , jan 0j < ". an ! 0., 2.19, 4.3.
, - - . 1.3 ab . 00 , 1+1 , 11 ,(+1)0 , 01 .
(xn) (yn) xn
yn.
2.19. xn > 0 n 2 N. xn ! x 2 R yn ! y 2 R xy , xn
yn ! xy . , xn ! 0 yn ! 1, xn
yn ! +1.. 1. 2.19, npa! 1 a > 0., (a) ( 1
n). a ! a 1
n! 0,
npa = a
1n ! a0 = 1.
2. (an) a > 1, 0 < a < 1 , , 2.19. (a) (n). a ! a n ! +1, an ! a+1 , +1, a > 1, 0, 0 < a < 1.
3. npn ! 1 2.19. (n)
( 1n), n! +1 1
n! 0, (+1)0 .
(+1)0 00 , - +1 0 0 0 [0;+1] .. 1. n! +1, 1
n! 0 n 1n = npn! 1.
2. a > 1, an ! +1, 1n! 0 (an) 1n = a! a.
3. a > 1, an ! +1, 1n! 0 (an) 1n = 1
a! 1
a.
4. nn ! +1, 1n! 0 (nn) 1n = n! +1.
5. nn ! +1, 1n! 0 (nn) 1n = 1
n! 0.
39
-
6. nn ! +1, (1)n1n
! 0 (nn) (1)n1n = n(1)
n1 . ,
n(1)n1
= n 1 n 2 N n(1)n1 = 1n 1
2 n 2 N.
7. , (xn) . xn ! 0, yn ! 0 (xnyn) [0;+1] .
2.19 01 = +1, , , . 01 .
. 1. 1n! 0, n! 1 ( 1
n)n = nn ! +1.
2. 1n! 0, 2n 1! 1 ( 1
n)2n1 = n2n+1 ! 1.
3. (1)n
n! 0, n ! 1 (1)n
n
n= (1)n2nn , (1)n2nn =
nn 4 n 2 N (1)n2nn = nn 1 n 2 N. , 1+1 11 , -
2.5.
.
1. 2.2, 2.3, 2.6, 2.9, 2.10, 2.11, 2.12, 2.13, 2.15,2.16 2.17.
2. (n+1)27(n+3)79
(2n+1)106
,n2+(1)nn+ 1
n
3n+2(1)n1pn,n(n+1)n+4
4n34n2+1
,
((1 n)5 + n4), ( n3+n+13n2+3n+1
)9,
3n+(2)n3n+1+2n+1
, (pn+ 1pn), (pn2 + n+ 1pn2 + 1).
3. - - (1 + 2+ 22 + + 2n), 1 + 12+ + 1
2n
,
(1 2 + 22 + + (1)n2n), 2737+ 2
8
38+ + 2n+6
3n+6
,2n
3n+ 2
n+1
3n+1+ + 22n
32n
.
4. 212+1
313+1
n1n+1
,23123+1
33133+1
n31n3+1
.
5. x 6= 1 xn 6= 1 n 2 N. xn ! x xn1xn ! x1x .6. (xn) -
: xn+1 = xn+2, xn+3 = xn3, xn+1 = xn23, xn+2 = xn2+3,xn+1 = xn
2 + 3, xn+2 = xn+1 + xn3 ;
7. (sinn);
8. (xn) :1 < xn n2+3nn2+1 , log10 n22 log10 n+4 < xn :+1; x > 00; x = 0
1; x < 0[nx] [ny]!
8>:+1; x > y0; x = y
1; x < y
nx [ny]
8>>>>>:! +1; x > y! 0; x = y 2 Z! 1; x < y ; x = y 2 Q n Z
, x = y 2 R nQ, 17 2.9.10. , . n5+4n3 < 100;
n7 35n6 + n3 47n < 84 n 2 N; 32< 7n
3n+54n3+n2+35
< 2;
2n4n3+7
n3+n2+3 78; 2n3n2+7n+1n3+n2+3
1 n 2 N;
11. 2.3, 2(1)
n1,
1 + (1)n12
n,(1)n1 + 10
n3
,(1)n1 n
n+1
.
12. [] jxnj ! 0, xn ! 0 : . , , :
xn ! 0 jxnj ! 0:
[] jxnj ! a > 0 xn ! x, x = a x = a. 2.18.
13. xn ! x yn ! y, x; y 2 R, maxfxn; yng ! maxfx; yg minfxn; yng ! minfx; yg.
14. : .
[] : 1 = n 1n= 1
n+ + 1
n! 0 + + 0 = 0. 1! 0.
[] : n = ( npn)n = n
pn npn! 1 1 = 1. n! 1.
2.11, 2.13, 2.19;
15. xn ! x 2 R yn ! y 2 R . x < y, xn < yn .16. , xn 2 [l; u] n 2 N xn ! x, x 2 [l; u].
x (xn), xn 2 (l; u) n 2 N; ;
17. . .
41
-
[] xn ! +1 (yn) , xn + yn ! +1.[] xn ! 1 (yn) , xn + yn ! 1.[] xn ! 0 (yn) , xnyn ! 0.[] xn ! +1 1 yn > l > 0, xnyn ! +1 1, .[] xn ! +1 1 yn < u < 0, xnyn ! 1 +1, .
18. .
[] b < 1 jxn+1j bjxnj, xn ! 0.: jxn+1j bjxnj n n0 , jxnj jxn0 jbn0 bn n n0 .[] b > 1 xn+1 bxn > 0, xn ! +1.[]
xn+1xn
! a < 1, xn ! 0.: [].
[] xn+1xn
! a > 1, xn ! +1 xn ! 1.[] a > 1, a
n
n! +1. , (n!)2
(2n)!! 0, 2nn!
nn! 0
an
n!! 0 a.
[] x 2 R (xn) xn+1xn ! 1 xn ! x.19. [] (xn), (yn) (xn + yn) .
[] (xn), (yn) (xnyn) .
20. [] (xn+ yn) (xn), (yn) , , , .
[] (xnyn) (xn), (yn) , , , .
21. (xn), (yn) xn; yn > 0 n 2 N, xn ! 0, yn ! +1 (xnyn) .
22. x1 > 0 xn+1 x1 + + xn n 2 N. 0 < a < 2, xnan! +1. (2n) a = 2.
23. . .
[] x (rn) rn 2 Q n 2 N rn ! x.: n 2 N rn 2 Q x 1n < rn < x+ 1n .[] x (tn) tn 2 R n Q n 2 N tn ! x.[] x (rn) (sn) rn; sn 2 Q n 2 N rn ! x sn ! x.
24. [] , , , Cesro: xn ! x 2 R, x1++xnn ! x . Cesro : , .
42
-
x 2 R. " > 0. n00 2 N jxn xj < "2 n 2 N, n n00 . n0 = max
n00;2(jx1xj++jxn001xj)
"
+ 1
. , n 2 N,
n n0 jx1++xnn xj = (x1x)++(xnx)
n
jx1xj++jxnxjn
=jx1xj++jxn001xj
n+
jxn00xj++jxnxjn
< "2+ (nn0
0+1)"2n
< "2+ "
2= ". x1++xn
n! x.
x = +1. M > 0. n00 2 N xn > 2(M + 1) n 2 N,n n00 . n0 = max
2n0
0 1; jx1 + + xn001j + 1. , n 2 N, n n0 x1++xnn =
x1++xn001n
+xn00++xn
n> 1 + 2(nn00+1)(M+1)
n
1 + (M + 1) =M . x1++xnn
! +1. x = 1, x = +1 (xn). 11 2.9 Cesro, . , Cesro: xn > 0 n 2 N xn ! x 2 [0;+1], npx1 xn ! x.
Cesro. , , ,, 4 4.3 Cesro .
[] xn = (1)n1 n 2 N, x1++xnn ! 0. , xn = 0 n 2 N xn = n n 2 N, x1++xnn ! +1 (xn) . Cesro.
[] Cesro: (xn), (yn) yn > 0 n 2 N y1 + + yn ! +1. xnyn ! l 2 R, x1++xny1++yn ! l.: Cesro.
25. [] xn ! x xn x n 2 N, supfxn jn 2 Ng = x.[] xn; x < y n 2 N xn ! x, supfxnjn 2 Ng < y.
26. xn ! x, inffxn jn 2 N; n kg x supfxn jn 2 N; n kg k 2 N.
27. [] xn ! x xn < x n 2 N, (xn) .[] xn ! x k 2 N xk x, (xn) .
28. jxn xmj 1 n;m 2 N, n 6= m. jxnj ! +1. (xn); (n), (n), ((1)n1n).
2.5 . e, .
2.1 . , , , , 1. - : ((1)n1) ((1)n1n) .
2.1. . :1. (xn) , limn!+1 xn = supfxn jn 2 Ng. : (xn) , +1, ,
43
-
.2. (xn) , limn!+1 xn = inffxn jn 2 Ng. : (xn) , 1, , .
. 1. - fxn jn 2 Ng. - - supremum , - - supremum +1. (xn) . supfxn jn 2 Ng = +1 xn ! +1. M > 0. M fxn jn 2 Ng, n0 2 N xn0 > M . (xn) , xn xn0 > M n 2 N,n n0 . xn ! +1. (xn) . x = supfxn jn 2 Ng xn ! x. " > 0. x " < x, x " fxn jn 2 Ng. n0 2 N x" < xn0 . (xn) , x" < xn0 xn n 2 N, n n0 . , x fxn jn 2 Ng, xn x < x+ " n 2 N. x " < xn < x+ " n 2 N, n n0 , , xn ! x.2. .
2.1. (xn) , , 2.1, (xn) , x, . , xn x n 2 N. , , (xn) , xn < xn+1 x , ,xn < x n 2 N. .:
(xn) xn ! x, xn x n 2 N. , , (xn) , xn < x n 2 N. (xn) xn ! x, xn x n 2 N. , , (xn) , xn > x n 2 N.
2.1 . . : , , , , n- . 2.1 (, ), .
2.1 . , - , - - . , , 6 2.4.
2.1 supremum R ,, . 2.1 , , supremum. , supremum . 12.
. (xn) x1 = 1 xn+1 =p2xn n 2 N.
(xn) 1;p2 ;p2p2 ; : : : .
44
-
. , x1 x2 xn xn+1 n 2 N. 0 1 . : xn xn+1 2xn 2xn+1
p2xn p2xn+1
xn+1 xn+2 . xn xn+1 n 2 N, (xn) , , . xn xn+1 xn
p2xn , xn 2
n 2 N. (xn) , , . xn ! x. xn+12 = 2xn n 2 N, x2 = 2x, x = 0 x = 2. (xn) x1 = 1, xn 1 n 2 N. x 1 ,, x = 2. (xn) . xn xn+1 xn
p2xn ( xn 0) xn 2. ,
xn 2 n 2 N, (xn) 2. . x1 2 . xn 2 n 2 N. xn+1 =
p2xn
p2 2 = 2,
xn 2 n 2 N. 2.20.
(1 + 1
n)n .
. (1+ 1n)n < (1+ 1
n+1)n+1 (n+1
n)n < (n+2
n+1)n+1
nn+1
(n+1n)n+1 < (n+2
n+1)n+1 n
n+1< ( n
2+2nn2+2n+1
)n+1 nn+1
< (1 1n2+2n+1
)n+1
Bernoulli. , n 2 N (1 1
n2+2n+1)n+1 > 1 n+1
n2+2n+1= 1 1
n+1= n
n+1:
(1 + 1n)n < 4 , , 1
2< p
npn+1
n n 2 N.
Bernoulli pnp
n+1
n=1
pn+1pnpn+1
n 1 npn+1pnpn+1
> 1 npn+1pnp
n
= 1pn(pn+ 1pn) = 1pnp
n+1+pn> 1
pn
2pn= 1
2
n 2 N.. 2.1 2.20,
(1 + 1
n)n.
e. ,
e = limn!+1(1 + 1n)n :
e
(1 + 1
n)n. e
(1 + 1n)n ! e.
4 (1 + 1
n)n.
(1+ 1n)n ! e (1+ x
n)n ! ex , 2
x 2 Z 5 2.7 15 2.9 x 2 R. 3.5.3 12 3.5.
((1 + 1n)n) , (1 + 1
n)n < e n 2 N.
e . 8.8.
. e , loge y, y > 0 log y ln y.
45
-
2.21 , , 1.6.
2.21. 1. log(yz) = log y + log z y; z > 0.2. log y
z= log y log z y; z > 0.
3. log(yz) = z log y y > 0 z.4. loga y =
log ylog a
y > 0 a > 0, a 6= 1.5. log 1 = 0, log e = 1.6. 0 < y < z, log y < log z.
1+1 11 , 1 +1 1 1 [0;+1] .. 1. 1! 1, n! +1 1n = 1! 1.2. a > 1 b = log a. 1 + 1
n! 1, bn! +1 (1 + 1
n)bn = ((1 + 1
n)n)b ! eb = a.
3. npn! 1, n! +1 ( npn)n = n! +1.
4. (xn) (yn) xn ! 1, yn ! +1 (xnyn) [0; 1].
5. 1npn n(1)n1
n npn n 2 N, n (1)n1n ! 1. , n ! +1
n(1)n1
n
n= n(1)
n1 , n(1)
n1= 1
n 1
2 n 2 N
n(1)n1
= n 1 n 2 N.6. (xn) (yn) xn ! 1, yn ! 1 (xn
yn) [0;+1] . .
. 1. (xn), xn = 1 + 11! + 12! + + 1n! n 2 N. (xn) , ,
1 + 11!+ 1
2!+ + 1
n!! e:
xn+1 xn = 1(n+1)! > 0 n 2 N, (xn) . k! 2k1 k 2 N. k = 1 , k 2 N, k 2 k! = 1 2 3 k 1 2 2 2 = 2k1 . ,
xn = 1 +11!+ 1
2!+ + 1
n! 1 + 1
20+ 1
21+ + 1
2n1 = 1 +1( 1
2)n
1 12
< 1 + 11 1
2
= 3
n 2 N. (xn) , , , , ., tn =
1+ 1
n
n n 2 N. Newton
9 1.3,
tn = 1 +n1
1n+n2
1n2
+ + nk
1nk
+ + nn
1nn
= 1 + 11!+ 1
2!(1 1
n) + + 1
k!(1 1
n)(1 2
n) (1 k1
n) + + 1
n!(1 1
n) (1 n1
n):
> 0 < 1, tn 1 + 11! + + 1n! = xn n 2 N. k; n 2 N, 1 k n, () k-,
tn 1 + 11! + 12!(1 1n) + + 1k!(1 1n)(1 2n) (1 k1n ):
46
-
n! +1, e 1 + 11!+ 1
2!+ + 1
k!= xk k 2 N ,
, e xn n 2 N. tn xn e n 2 N , tn ! e, xn ! e. 8. , 10, 1 + 1
1!+ 1
2!+ + 1
n!! e 1 + x
1!+ x
2
2!+ + xn
n!! ex x.
2. (xn), xn = 1 +12+ 1
3+ + 1
n n 2 N.
1 + 12+ 1
3+ + 1
n! +1:
xn+1 xn = 1n+1 > 0 n 2 N, (xn) . x2n xn = 1n+1 + + 1n+n 1n+n + + 1n+n = nn+n = 12
n 2 N. ,x2 x1 12 ; x22 x2 12 ; x23 x22 12 ; : : : ; x2k1 x2k2 12 ; x2k x2k1 12 :
, x2k x1 k2 , , x2k k2 + 1 k 2 N. (xn) , (xn) , xn ! +1. 6 2.7 2 2.8.
3.
1 + 1
22+ 1
32+ + 1
n2
:
xn = 1+122+ 1
32+ + 1
n2 n 2 N. xn+1 xn = 1(n+1)2 > 0 n 2 N,
(xn) .
xn 1 + 112 + 123 + + 1(n1)n = 1 + 11 12 + 12 13 + + 1n1 1n = 2 1n < 2 n 2 N. (xn) , , . 2.8.
. (an) (bn) an bn n 2 N. , [a1; b1], [a2; b2]; : : : [an+1; bn+1] [an; bn] n 2 N. :(i) (an) (bn) .(ii) x an x bn n 2 N.(iii) x (ii) bn an ! 0. , (an), (bn) x .
. (bn) , an bn b1 n 2 N, (an) , , . (an) an ! a., (an) , a1 an bn n 2 N, (bn) , , . (bn) bn ! b. an bn n 2 N, a b. an a b bn n 2 N. x 2 [a; b] an a x b bn n 2 N. , x an x bn n 2 N, a x b, x 2 [a; b]. x an x bn n 2 N [a; b]. , x a = b , ,bn an ! 0. x x = a = b.
47
-
(ii) , , 13.
. x 2 [0; 1) p 2 N, p 2. n 2 N xn = [pnx] p[pn1x]. (xn) p- x., n 2 N sn = x1p + + xnpn tn = x1p + + xnpn + 1pn . (sn) p- ( ) x (tn) p- x.
xn 2 Z n 2 N. [pn1x] pn1x < [pn1x] + 1 p[pn1x] pnx < p[pn1x] + p p[pn1x] [pnx] < p[pn1x] + p, 0 xn < p , xn 2 Z, 0 xn p 1., n 2 N xn 0; 1; : : : ; p 1. sn+1 sn = xn+1pn+1 0 tn+1 tn = xn+1pn+1 + 1pn+1 1pn p1pn+1 + 1pn+1 1pn = 0 n 2 N. (sn) (tn) . , sn tn n 2 N, ., tn sn = 1pn ! 0, (sn), (tn) . ;
sn =[px]p [x]+ [p2x]
p2 [px]
p
+ + [pn1x]
pn1 [pn2x]pn2
+[pnx]pn
[pn1x]pn1
= [p
nx]pn
[x] = [pnx]pn
:
, [pnx] pnx < [pnx] + 1 sn x < sn + 1pn = tn , sn ! x tn ! x.
p- x 2 [0; 1) : p1. - - n0 2 N xn = p1 n 2 N, n n0 . tn+1 = tn + xn+1pn+1 + 1pn+1 1pn = tn + p1pn+1 + 1pn+1 1pn = tn n 2 N, n n0 . (tn) , tn ! x, tn = x. sn x < tn n 2 N.
: p = 2 0; 1, p = 3 0; 1; 2 , , p = 10 0; 1; : : : ; 9.
p- 8.3.
.
1. 2.1.
2. (1 + 1n)n ! e.
[] : (1 + 1n)n+3 ! e, (1 + 1
n+2)3n+5 ! e3 , (1 1
n)n ! 1
e, (1 + 3
n)n ! e3 ,
(1 3n)n ! 1
e3.
: 1 + 3n= (1 + 1
n)(1 + 1
n+1)(1 + 1
n+2).
[] (1 + kn)n ! ek k 2 Z.
5 2.7 k. k 2 Q k 2 R.
3. [] , Bernoulli, (1 + 1
n)n+1
, e (1 + 1n)n+1 > e n 2 N.
[] (k+1k)k < e < (k+1
k)k+1 k = 1; 2; : : : ; n 1
nn
n!< en1 < n
n+1
n! n 2 N.
48
-
npn!n! 1
e.
19 7.3.
npn!! +1:
8.4, .
4. (1 + 1
n)n.
(1 + 1
n)n . , Bernoulli,
(1 + 1
n)n+1
. , ,
, .
5. .
[] x1 = 1 xn+1 = xn +1
xn2 n 2 N. (xn)
.
[] 7xn+1 = xn3 + 6 n 2 N. , x1 ,
(xn) .
[] x1 > 0 xn+1 =6+6xn7+xn
n 2 N. , x1 , (xn) .
[] xn+1 = 2 1xn n 2 N. x1 = 1, (xn) . x1 > 1, (xn) . x1 < 1 x1 6= k1k k 2 N, (xn) . x1 =
k1k
k 2 N;[] xn+1 = sinxn n 2 N. (xn) , , .
[] x1; x2 > 0 xn+2 = xn+1 + xn n 2 N. (xn+1
xn) .
6. .
[] a > 1, (an) , an+1 = aan , an ! +1. 0 < a < 1.[] a > 1, (a
n
n) ,
an+1
n+1= an
n+1an
n, a
n
n! +1.
[] a > 1, ( npa) ,
2npa2
= npa , n
pa ! 1. a = 1,
0 < a < 1;
[] ( npn)
npn! 1.
[] (an
n!) , a
n+1
(n+1)!=
an
n!a
n+1, a
n
n!! 0.
7. 1.2 xk = a.
a > 0 k 2 N, k 2. (xn) x1 > 0 xn+1 =
k1kxn +
1k
axnk1 n 2 N. xn > 0
n 2 N. , Bernoulli, xnk a n 2 N, n 2 , , xn+1 xn n 2 N, n 2. (xn) , x = limn!+1 xn , x > 0 xk = a.
49
-
8. . (xn) xn+1 xn+xn+22 n 2 N. n 2 N, .[] (xn). (xn xn+1) xn xn+1 ! 0. (xn) . (xn). (xn) ;
[] .
9. [] 0 < x1 y1 xn+1 = pxnyn , yn+1 = xn+yn2 n 2 N, (xn) , (yn) , xn yn n 2 N (xn), (yn) , - x1, y1.
[] 0 < x1 y1 xn+1 = 2xnynxn+yn , yn+1 =pxnyn n 2 N, (xn)
, (yn) , xn yn n 2 N (xn), (yn) , - x1, y1.
[] - - .
10. xn =135(2n1)246(2n) n 2 N. (nxn2)
((n + 12)xn
2) . . , , 11 7.3.
11. 1.5.
[] f : [a; b]! R [a; b]. f(a) > a f(b) < b, 2 (a; b) f() = .: f(x) 6= x x 2 [a; b]. f(a+b
2) < a+b
2, a1 = a; b1 =
a+b2
, f(a+b2) > a+b
2, a1 =
a+b2
, b1 = b. : f(a1) > a1 f(b1) < b1. , [a1; b1], [a2; b2]; : : : [an+1; bn+1] [an; bn], f(an) > an, f(bn) < bn bn an = ba2n n 2 N. an bn n 2 N an ! , bn ! . f() < f() > . .[] I f : I ! R x 2 I 0 > 0 f(x0) f(x) f(x00) x0; x00 2 (x 0; x+ 0) \ I , x0 < x < x00 . f I .
: a; b 2 I a < b, f(a) > f(b). f(a) > f(a+b2), a1 =
a; b1 =a+b2
, f(a+b2) > f(b), a1 =
a+b2
, b1 = b. f(a1) > f(b1). , [a1; b1]; [a2; b2]; : : : [an+1; bn+1] [an; bn], f(an) > f(bn) bn an = ba2n n 2 N. an bn n 2 N an ! , bn ! . 0 , .
12. . , supremum.
. , 1
2n! 0. , - A. x1 2 A
y1 A. [x1; y1] A A. x1+y1
2 A, x2 = x1, y2 =
x1+y12
, , x2 =x1+y1
2,
y2 = y1 . [x2; y2] A A. ,
50
-
[x1; y1], [x2; y2]; : : : [xn+1; yn+1] [xn; yn] yn xn = y1x12n1 n 2 N [xn; yn] an 2 A un A. u xn ! u, yn ! u , , an ! u, un ! u. u A.
13. (ii) .
(an) (bn) an bn n 2 N. fan jn 2 Ng, fbn jn 2 Ng , x an x bn n 2 N.
14. A - R - - a : N ! A -- . an = a(n), A - A = fan jn 2 Ng , , A . , A - .
, I ( ) .
- - I = fan jn 2 Ng. [x1; y1] I y1x1 > 0 a1 /2 [x1; y1]. [x2; y2] [x1; y1] y2x2 > 0 a2 /2 [x2; y2]. , [x1; y1],[x2; y2]; : : : [xn+1; yn+1] [xn; yn] an /2 [xn; yn] n 2 N. 2 [xn; yn] n 2 N , , 6= an n 2 N. .
15. .
1 , n 2 N, n 2, 2n 2n . pn qn , , .
[] p2 = 4p2, q2 = 8
pn+1 = 2pn2 +
4 pn2
4n
12 1
2 qn+1 = 4qn2 +
4 + qn
2
4n
121
n 2 N, n 2.[] qn = pn
1 pn2
4n+1
12 n 2 N, n 2.
[] (pn) , (qn) pn < qn n 2 N, n 2.[] (pn), (qn) .
, , , 2, : pn 2 qn n 2 N, n 2. pn ! 2 qn ! 2.
2.6 Supremum, inmum .
A. (xn) A xn 2 A n 2 N, (xn) A.
2.22 supremum - inmum - : supA A infA A.
51
-
2.22. - A.1. A supA A supA.2. A infA A infA.
. 1. A , x = supA . n 2 N x 1
n A, xn 2 A x 1n < xn x.
(xn) A xn ! x., A , supA = +1. n 2 N A, n 2 N xn 2 A xn > n. (xn) A xn ! +1., (xn) A. (xn) , supA, xn supA n 2 N. A supA.2. .
supremum inmum 4.
.
1. 2.22.
2. [0; 2], [0; 2), f2g, [0; 1] [ f2g supremum . , ( ) , , . ,, , .
3. N, Z, Q, f 1njn 2 Ng, fx 2 R n Q j 0 < x 1g, fx 2 Q j 0 < x < p2g
, supremum inmum .
4. - A u A. u = supA A u. infA l A.
5. - A. supA 2 A A supA. supA /2 A, A supA. infA.
2.7 .
. (xn). n1 , n2 , n3 , : : : , nk , : : : n n1 < n2 < < nk < nk+1 < . (xn). x1 , x2 , : : : , xn , : : : xn1 ; xn2 ; : : : ; xnk ; : : : . : xn1 , xn2 . , (xnk). , (xnk) (xn).
52
-
, (xnk) x n : N ! R (x n)(k) = x(n(k)) = xn(k) = xnk k 2 N, n : N! N x : N! R, , , (nk) (xn).
, n1 < n2 < < nk < nk+1 < , .
. 1. n1 = 2, n2 = 5, n3 = 6, n4 = 9, n5 = 13, (xn) x2 ; x5 ; x6 ; x9 ; x13 .
2. , n1 = 2, n2 = 5, n3 = 6, n4 = 10, n5 = 8 (xn). x2 ; x5 ; x6 ; x10 ; x8 (xn) : x10 x8 (xn) - x9 - x10 x8 .
. .1. nk = 2k k 2 N (x2k) (x2; x4; x6; x8; x10; : : : ) (xn).
2. nk = 2k1 k 2 N (x2k1) (x1; x3; x5; x7; x9; : : : ) (xn).
3. nk = k k 2 N (xk) (x1; x2; x3; x4; x5; : : : ), (xn). , (xn) (xn).
4. nk = 2k1 k 2 N (x2k1) (x1; x2; x4; x8; x16; : : : ).
5. nk = k2 k 2 N (xk2) (x1; x4; x9; x16; x25; : : : ).
(xnk) k. k 1; 2; 3; : : : , nk (xn).
2.2. nk 2 N nk < nk+1 k 2 N. nk k k 2 N.. n1 1 n1 2 N. nk k k 2 N. nk+1 > nk nk; nk+1 2 N, nk+1 nk + 1, nk+1 k + 1. nk k k 2 N. 2.23. , .
. xn ! x 2 R (xnk) (xn). xnk ! x. " > 0. n0 2 N xn 2 Nx(") n 2 N, n n0 . , k 2 N, k n0 nk n0 , , 2.2, nk k. k 2 N, k n0 xnk 2 Nx("). xnk ! x.
2.23 , , : , . - 16 13 2.9.
. ((1)n1) . , (1)(2k1)1 = 1 ! 1 (1)(2k)1 = 1! 1.
2.24 .
53
-
2.24. x 2 R x2k ! x, x2k1 ! x. xn ! x.. " > 0. k0
0 2 N x2k 2 Nx(") k 2 N,k k00 . , k000 2 N x2k1 2 Nx(") k 2 N, k k000 . n0 = maxf2k00; 2k000 1g. n0 2 N xn 2 Nx(") n 2 N, n n0 .
, 2.24 8 9.
.
.
1 1
2+ 1
3 1
4+ + (1)n1 1
n
:
xn = 1 12 + 13 14 + + (1)n1 1n n 2 N. x2k+2 x2k = 12k+1 12k+2 > 0 k 2 N. ,
x2k = 1 (12 13) (14 15) ( 12k2 12k1) 12k < 1
k 2 N, . (x2k) , , ., x2k+1 x2k1 = 12k
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