m16ลำดับอนุกรม.pdf

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Math Online VI http://www.pec9.com !""#$ %6 &'()*!+&,-./012

1

!"#$%&'$( ) *++,- .6 /0&12*3/45678(9

!"#$%& ' ()*+,-

./0*12*345678,

{ (1 , 5) , (2 , 7) , (3 , 9) , (4 , 11) , }

!"#$%#&'#&()*+,-*./01%231-$(!"#$%#45%67'%8%#9:$;8* N 67'%8%8'$?0$@0%A.BCDE"9=DF4%(G#45%H7'"0;1. { (1,1) , (3,2) , (5,3) , (7,4) , }

2. { (1,1) , (2,3) , (3,5) , (4,7) , }

3. { (1,1) , (1,2) , (1,3) , (1,4) , }

4. { (1,1) , (5,2) , (7,3) , (9,4) , } (2)

9/:%$)*

9/:%6*1;

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2

()*+,-;(2/!H7'"0;#HB>ML9>PDH7'"0;231-#$P1D%7'@6%.B8'$PD90G--K(1%(G#&()*8='RH9='-&=8$(d)./0*12*!,9"=?*5 5 , 7 , 9 , 11 ,

7 5 9 7 11 9

= 2 = 2 = 2

RH9='-&=8$(d)

#&'?'$'&ST'@6%.K018F4(an) F"C6'*?$*'&an = a1+ (n 1) d

#$P1D a1= @6%.K(11 BD-H7'"0; n = H7'"0;K(1BD-@6%.K(16JT' d = RH9='-&=8$@1A" an = d n + (a1 d)

4. 6-T'@6%.K018F4BD-H7'"0;#HB>ML99=DF4%(G5 , 7 , 9 , 11 , (2n+3)

9/:%$)*

5. 6-T'@6%.K018F4BD-H7'"0;#HB>ML94 , 7 , 10 , . ( 3n + ')

9/:%$)*

6. 6-T'@6%.K(1 n (@6%.K018F4) BD-H7'"0;#HB>ML95 , 9 , 13 , . (4n + ')

9/:%$)*

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3

7. 6-T'@6%.K(1 30 BD-H7'"0;#HB>ML99=DF4%(G2 , 6 , 10 , 14 , ( ''8 )

9/:%$)*

8. 6-T'@6%.K(1 30 BD-H7'"0;#HB>ML99=DF4%(G2 , 4 , 10 , . ('72)

9/:%$)*

9. 6-T'@6%.K(1 100 BD-H7'"0;#HB>ML9

x , 2x , 5x , . (296x)

9/:%$)*

'0. 176 #45%@6%.K(1#K='E"BD-H7'"0;#HB>ML91 , 6 , 11 , . ( ;BC#.0#7$%& 36 )

9/:%$)*

''. 6-T'67'%8%K(1D)Q=&JT8='- 6 ML96-T'D(* 5 @6%.231-#&()-D)Q=&JT8='-@6%.K0G-?D-K(1*7'T%"ETC%(G (9 , '3 , '7 , 2')9/:%$)*

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4

'3. *7'T%"@6%.K(1 4 ML9#45% 12 ML9 SC'RH;8*BD-#HB 4 67'%8%%(G#45% 28

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5

'6. &JT8='- 100

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6

'9. 67'%8%?$'/L*E%#29 { 100 , 101 , 102 , , 600} 231-T'&"C8) 8 T&PD 12 H-908#K='*0;BCDE"

1. 84 2. 92 3. 100 4. 125 (='>-K(1#?$D>='>-K(1%(G#&()*8='D09&'?=8%&=8$( r )./0*12*!,9"=?*5 2 , 4 , 8 , 16 ,

42 84 168

= 2 = 2 = 2

D09&'?=8%&=8$(r)

#&'?'$'&ST'@6%.K018F4(an) F"C6'*?$*'&

an = a1 . r(n 1)

#$P1D a1 = @6%.K(11 BD-H7'"0;n = H7'"0;K(1BD-@6%.K(16JT'r = D09&'?=8%&=8$

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7

20. 6-T'@6%.K018F4BD-H7'"0;#&B'>ML99=DF4%(G1 , 3 , 9 , 27 , (an= 3

(n ') )

9/:%$)*

2'. 6-T'@6%.K018F4BD-H7'"0;#&B'>ML9121 ,

61 ,

31 , . (

24

n2 )

9/:%$)*

22. 6-T'@6%.K(1 9 BD-H7'"0;#&B'>ML9 1 , 2 , 4 , 8 , ( 256 )9/:%$)*

23. 6-T'@6%.K(1 10 BD-H7'"0;#&B'>ML9 3 , 6 , 12 , . ('536)

9/:%$)*

24. 6-T'@6%.K(1 7 BD-H7'"0;#&B'>ML99=DF4%(G 3 , 6 , 12 , . ('92)

9/:%$)*

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8

25. 162 #45%@6%.K(1#K='E"BD-H7'"0;#&B'>ML9 2 , 6 , 18 , . (.0#7$%&5)9/:%$)*

26. SC' 5 ML96-T'67'%8%K(1D)Q=&JT8='-67'%8%K0G-?D-%(GML9 ('0 D(E '0)

9/:%$)*

27. #$P1D67'%8%67'%8%T%31-F4;8**0;

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9

28. RH;8*BD-H7'"0;#&B'>ML9 3 @6%.$(>='#K='*0;221 QMBD-@6%.K0G-?'$$(>='#K='*0;

27 ='$'*K(1?U"$(>='#K='E" ( 6 )9/:%$)*

29. RH;8*BD-#HB 3 67'%8%K(1#&()-*0%#45%H7'"0;#&B'>ML9$(>='#K='*0; 19 QMBD-#HB?'$67'%8%%(G#45% 216 6-T'67'%8%K(1$(>='%CD)K(1?U" ( 4 )

9/:%$)*

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10

!"#$%& 2 (/G/!='6J#@L1$B3G%#&P1D)V9'$

#$P1D n #BC'E*HC >='an 6J;D*F$=F"C8='$(>='#K='E"*&M(#/=%%(G6JT'>='HL$L9F$=F"C

nalimn

= T'>='F$=F"C

H7'"0;K(1T'>='HL$L9F$=F"C#&()*H7'"0;F"#8D&.#6%9.H7'"0;K(1T'>='HL$L9F"C #&()*H7'"0;>D%#8D&.#6%9.

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11

./0*12*()*+,-!?"HB#%Ian = (1)

n

6JF"C8=' a1 = (1)1 = 1a2 = (1)

2 = 1

a3 = (1)3 = 1

a4 = (1)4 = 1

a5 = (1)5 = 1

a6 = (1)6 = 1

.

a = ?

H7'"0;K(1T'>='

limn a

n

!"#

$%&'(

)*+",-)./$0#+(convergent sequence )

H7'"0;K(1T'>=' limn

an !1/!"# $%&'( )*+",-)./22((divergent sequence )

H7'"0;)./22(3&41&5/+6),-!71+ $%&'( )*+",-8(9/:(9,"(ossilating sequence )

9/:%6*1@*>?*(/G/!

9/:%$%&' E/C?$;09LBD-HL$L9-K(1

#/=% limn

5 = 5 limn

4 = 4 limn

0 = 0

2) limn

(anbn) = limn

an limn

bn

3) limn

(an. bn) = limn

an . limn

bn

4)limn nb

na =nblimn

nalimn

5) limn

(an)k = lim

n an

k

6) limn

kna = k nalim

n

6J#T:%8='>='K(1F"C6J#45%+1 ?H0;*0;1 F4#&P1D)V#$P1Dn #BC'E*HC >=' an 6J;D*F$=F"C8='#45%+1 T&PD 1 *&M(#/=%%(G6JT'>='HL$L9F$=F"C

nalimn

= T'>='F$=F"C

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12

7) limn

| an| = | limn

an|

8) limn

(log an) = log limn

an

9) limn

(sin an) = sin limn

an

0 #$P1D| a | < | b | #$P1D| a | > | b |

11) limn

[ kn1 ] = 0 #$P1D k R+

12) limn

[ nb1 ] = 0 #$P1D b 0

30. 6-#9L$>7'H-E%/=D-8='-9=DF4%(G1) lim

n c =

..................................

2) limn

(anbn) = ..................................

3) limn

(an. bn) = ..................................

4) limn n

bna =

..................................

5) limn

(an)k =

..................................

6) limn

kna = ..................................

7) limn

| an| = ..................................

8) limn

(log an) = ..................................

9) limn

(sin an) = ..................................

;0)n

lim [ba ]n =

11) limn

[kn1 ] =

..................................

12) limn

[ nb1 ] =

..................................

10)n

lim [ba ]n =

#/=% limn

[ 75 ]n = 0

limn

[49 ]n =

#/=% limn

[ 5n1 ] = 0

limn

[ n81 ] = 0

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13

!,9"=?*5$%&' 6-T'HL$L9BD-H7'"0; an= )12n3()12n2n6(

9/:%$)* 6'*n

lim an =n

lim

12n3

12n2n6

=n

lim

2n

12

n

23n

2n

1

2n

2n

2n

26n

=n

lim

2n13

2n1

n26

=

2

n

1lim

n

3lim

n

2n

1limn

n2

limn

6limn

=03

006

nlim an = 2

3'. 6-T'HL$L9BD-H7'"0; an= )22(4n)15n2(8n

(2)

9/:%$)*

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14

!,9"=?*5$%& 2 6-T'HL$L9BD-H7'"0; an = )13n3()12n2n6(

9/:%$)* 6'*n

lim an =n

lim

13

3n

12n2

n6

=n

lim

3n1

3n

33n3n13n2n3n

26n

=n

lim

3n13

3n1

2n2

n6

=

3n

1limn3lim

n

3n

1limn

2n

2lim

nn6

limn

= 03 000

nlim an = 0

32. 6-T'HL$L9BD-H7'"0; an= )24(5n)15n2(8n

(0)

9/:%$)*

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15

!,9"=?*5$%& 3 6-T'HL$L9BD-H7'"0; an = )1n3()13n2n4(

9/:%$)* 6'*n

lim an =n

lim

13n

13n2

n4

=n

lim

2n1

2n3n

2n12n3n2n 24n

=n

lim

2n1

n3

2n1

n34

=

2n

1lim

n

n3lim

n

2n

1limn

n3

limn

4limn

=00

004

nlim an = T'>='F$=F"C

33. 6-T'HL$L9BD-H7'"0; an = )1n()12n25(

n ()

9/:%$)*

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16

34(En 33) 6-T'>='BD- limn

1n51n2

1n31n5

(25)

9/:%$)*

6*1@*>?*(/G/!J+=9/:%(,+?7'T&0;+,-*./01%@TU%'$ ETC@L6'&M'#I@'J#KD$BD- n K(1$(*7'H0-?Q-?U"BD-#WOPD 2

nlim

13

3n

12n2

n6

= 0

*&M(K(1 2 T'**7'H0-?Q-?U"BD-#WO= *7'H0-?Q-?U"BD-?=8%

#/=% *7'H0-?Q-?U">PD 3 >7'9D;BD-HL$L9%0G% =LX?=8%?0$4&J?LKA

LX#WO?0$4&J?LKA

nlim

13

3n

12

2n3

n5

=35

*&M(K(1 3 T'**7'H0-?Q-?U"BD-#WO > *7'H0-?Q-?U"BD-?=8%#/=% *7'H0-?Q-?U">PD 2 >7'9D;BD-HL$L9%0G% 6JT'>='F$=F"C

nlim

13n

12n2

n6

= T'>='F$=F"C

*7'H0-?Q-?U">PD3

*7'H0-?Q-?U">PD3

*7'H0-?Q-?U">PD1

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17

35. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F4"0-9=DF4%(G

1. an = 123n427n

2. an = 98n23n85n22n

3. an = 53n2nn

4. an = 12n85n2n

5. an = 132n135n

!"- 1.37 2.

32 3. 0 4. T'>='F$=F"C 5.

25

36. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F4"0-9=DF4%(G

an =3

54n

1n3

(6427 )

9/:%$)*

37. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F4"0-9=DF4%(G

cn =

5

3n232n

4n2n

(2' )

38. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F49=DF4%(G

an = 12n 2n 12n 2n ( 2' )9/:%$)*

39. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F49=DF4%(Gan = 3n

2+ 4n (@*>?*HG?H+F)

9/:%$)*

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18

40. HL$L9BD-H7'"0; an =2n34 n5

13 nn

#K='*0;#K='E"

1. 51 2.

231 3. 0 4. 1 (?*(/G/!J+=9/:%(,+

?7'T&0;+,-*./01%#D*!4PD0#/=% Y'%?Q-?U">PD 3

nlim

1)n

7(5

n2)

n2(3

= 0

Y'%?Q-?U">PD5

*&M(K(1 2 T'*Y'%?Q-?U"BD-#WO= Y'%?Q-?U"BD-?=8%#/=% Y'%?Q-?U">PD 3 >7'9D;BD-HL$L9%0G% =

LX?=8%?0$4&J?LKALX#WO?0$4&J?LKA

nlim

1)n

7(3

n2)

n2(3

= 72

*&M(K(1 3 T'*Y'%?Q-?U"BD-#WO > Y'%?Q-?U"BD-?=8%#/=% Y'%?Q-?U">PD 5 >7'9D;BD-HL$L9%0G% 6JT'>='F$=F"C

nlim

1n

7(4

n2)

n2(5

= T'>='F$=F"C

Y'%?Q-?U">PD3

Y'%?Q-?U">PD

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19

42. 6-T'HL$L9BD-H7'"0;K(1$(@6%.K018F4"0-9=DF4%(G

1.n

lim

1)n

7(5

n2)

n2(3

2.n

lim

1)n

7(3

n2)

n2(3

3. n lim

1n7(4

n2)

n2(5

( 0 , 7

2 , @*>?*HG?H+F)

43(En 4') SC' an = 123n1n2n

='BD- limn

n51)(n

)n5(12n

(2)

9/:%$)*

45(G845) *7'T%"H7'"0; an= 32n5n2

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20

46. *7'T%"ETC A =n

lim n 3 PD>='E%BCDE"

1.41 2.

161 3.

21 4. 1 (

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21

49. *7'T%"H7'"0; an = 13n122n

HL$L9BD-H7'"0; an >PD >='E%BCDE"

1.32

2. 32

3. 0 4. F$=$(HL$L9 (ML9 4 , 7 , 10 , 13 , .... >PD ...... (an = 3n + ')B) HL$L9BD-H7'"0;42 ,

74 ,

106 ,

138 , .... >PD .......... (

32 )

9/:%$)*

5'(G836) 6-T'HL$L9BD-H7'"0;21

5x3 ,227x5 ,

239x7 , . (4 )

9/:%$)*

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22

!"#$%& 3 "#K61G

D%U*&$ (S) >PD RH;8*BD-?$'/L*BD-H7'"0;#/=% S5 = a1+ a2 + a3 + a4 + a5

S4 = a1 + a2 + a3 + a4S3 = a1 + a2 + a3 D%U*&$67'*0"S2 = a1 + a2

S1 = a1

S

= So = a1+ a2 + a3 + a4 + a5+ . D%U*&$D%0%9.

52. D%U*&$67'*0" >PD...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......D%U*&$D%0%9. >PD...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ....

D%U*&$

D%U*&$67'*0" D%U*&$D%0%9.

D.67'*0"#HB>ML9 D.67'*0"#&B'>ML9 D.D%0%9.#HB>ML9 D.D%0%9.#&B'>ML9

53. 6-#9L$

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23

54. 6-T'RH;8* 10 @6%.='#K='*0;BCDE"*. 890 B. 980 >. 1010 -. 1100 (

7/24/2019 m16 .pdf

24/38


24

59. D%U*&$#HB>ML9T%31-$( 10 @6%. RH;8*BD- 5 @6%.

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25

"#K61G0)*6,+;12/!D%U*&$67'*0"#&B'>ML9 >PD D%U*&$K(1#*L"6'*H7'"0;#&B'>ML967'*0"#&'?'$'&ST'RH;8*F"C!")D'W0)?$*'&

Sn = 1r

)1n(r1

a

T&PD Sn = 1r1

arn

a

#$P1D Sn >PD RH;8* n @6%.PD 67'%8%@6%.K(1;8*r >PD D09&'?=8%&=8$ a1 >PD @6%.PD @6%.?U"KC')

63. 6-T'RH;8* 7 @6%.

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26

66. D%U*&$#&B'>ML9/U"T%31- $(RH;8*BD- 3 @6%.='#45% 9 #K='BD-RH;8* 3 @6%.

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27

!"#$%&!"'"()

*'&T'>='D%U*&$D%0%9.D'6K7'F"C!")8LA("0-%(G9/:%$%&' T'>='6'* S0=

nlim Sn

9/:%$%&2 E/CTH0**'&9=DF4%(G"#K61G"#,#!7;(2/!D%U*&$D%0%9.#HB>ML9 >PD D%U*&$K(1#*L"6'*H7'"0;#HB>ML9D%0%9.

#/=% 3 + 7 + 11 + 15 + 19 + ...D%U*&$='$'**8=' 1 63-T'>='RH;8*D%0%9.F$=F"C

*&M(K(1 2 T'* r < 1

*&M(%(G 6J?'$'&ST'RH;8*F"C !")E/C?Q9& So = r11

a

D%U*&$D%0%9.K(1?'$'&ST'>='F"C #&()* "#K61G(L?;

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28

7'(G835) D%U*&$ 0.1 + 0.001 + 0.00001 + .... $(RH;8*#K='*0;................. (0.'0')9/:%$)*

72(G840) *7'T%"ETC a1+ a2+ + an+ #45%D%U*&$#&B'>ML9231-$( a2 = 59

='#K='*0;>='E%BCDE"1. 5 2. $(>='#*PD;6J#K='*0; 5 (

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29

75(En 22) *7'T%"ETC?'$#TH(1)$"C'%#K='&Q4K(1 n + 1 #*L"6'**'&H'*#?C%9=D6U"*31-*H'-"C'%K0G-?'$BD-?'$#TH(1)$"C'%#K='&Q4K(1 n SC'B;8%*'&?&C'-&Q4?'$#TH(1)$"C'%#K='%(GF$=?LG%?U"8'$?Q-#"L$E%&0G- 6-T'&J)JK(1#>HP1D%K(1F"CK0G-T$"E%

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30

D%U*&$

D%U*&$67'*0" D%U*&$D%0%9.

D.67'*0"#HB>ML9 D.67'*0"#&B'>ML9 D.D%0%9.#HB>ML9 D.D%0%9.#&B'>ML9

Sn= 2n [a1+ an] Sn= 1r

)1nr(1

a

T'RH;8*F$=F"C T'* r 1

Sn= 2n [2a1+ (n 1)d] Sn= 1r

1ar

n.a

T'RH;8*F$=F"C

T'* r 1

So = r11

a

!"#$%& 4 "#K61G(,6M2E./;NM

"#K61G1LB1)n.(n

1

77(G836) #&'K&';8='1)n.(n

1

= n1

1n1

( 71 )

6-T'RH;8*BD-D%U*&$87

1

+98

1

+109

1

+

9/:%$)*

78(G837) #&'K&';8='1)n.(n

1

= n1

1n1

"0-%0G%D%U*&$

1n 1)n.(n3

n

2

5

$(RH;8*#K='*0;>='E%BCDE"

1. 0 2. 2 3. 4 4. 5 (

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31

79. #&'K&';8=' ;kk;kk

;

6-T'>='BD-

1 + ;2;

+23

;

+34

;

+ . +;-nn

;

+. (@*>?*HG?H+F)

9/:%$)*

"#K61GOPG;(2/!+ ;12/!

80. RH;8*S3-D%0%9.BD-D%U*&$31 +

94 +

277 +

8110 +

$(>='9&-*0;>='E%BCDE"1.

45 2.

27 3. 4 4. 5 (

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32

8'. RH;8*BD-D%U*&$D%0%9.%(G $(>='#K='E" 3 +45 +

169 +

6417 + ...

1.3

16 2.4

18 3. 5 4. 6 (

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33

!"#$%& 5 P,QQ(,6M27D$#6*1-96

./0*12*P,QQ(,6M27!?"HB#%I

5

1i3)(i = (1 + 3) + (2 + 3) + (3 + 3) + (4 + 3) + ( 5 + 3)

= 4 + 5 + 6 + 7 + 8

= 30

4

1i)

2(i = 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16

= 30

5

3i

1)(2i = ( 2[3] 1) + (2 [4] 1) + (2 [5] 1)

= 5 + 7 + 9 = 20

84. 6-T'>='BD- (1)

7

1ii3 (2)

4

1i1)i(2 (8', '6)

9/:%$)*

PL!1P)*@1,-@*>?* Sumation

1 )

n

1iC = n C #$P1D C #45%>='>-908 #/=%

10

1i5 = 10 (5) = 50

2)

n

1iC ai = C

n

1ii

a #/=%

n

1i

2i5 = 5

n

1i

2i

3)

n

1i)

ib

i(a =

n

1i

ibn

1i

ia

4)

n

1ii =

21)(nn

#/=%

9

1ii =

21)(99

= 45

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34

5)

n

1i

2i =

61)(2n1)(nn

#/=%

6

1i

2i =

61)(2[6]1)(66

= 91

6)

n

1i

3i = 2

21)(nn

#/=%

8

1ii = 2

21)(88

= 1296

85. 6-#9L$>7'H-E%/=D-8='-9=DF4%(G

1 ) [

n

1iC =

........ ........ ........ ........ .. #/=%

10

1i5 =

........ ........ ........ ........

2) [

n

1iC ai = ........ ........ ........ ........ #/=%

n

1i

2i5 =

........ ........ ........ ..

3)

n

1i)

ib

i(a =

........ ........ ........ ..

86. 6-#9L$>7'H-E%/=D-8='-9=DF4%(G

1)

n

1ii =

........ ........ ........ ........ #/=%

9

1ii =

........ ........ ........ ........ ......

2)

n

1i

2i =

........ ........ ........ ...... #/=%

6

1i

2i =

..........................................

3)

n

1i

3i =

........ ........ ........ ...... #/=%

8

1i

3i =

........ ........ ........ ........ ........

87. 6-T'>='BD-

10

1i25i ('925)

9/:%$)*

88. 6-T'>='BD-

10

1i8)

3(i (3'05)

9/:%$)*

7/24/2019 m16 .pdf

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35

89. 6-T'>='BD-

10

i5)(3i

5

('65)

9/:%$)*

90. SC'

7

1n2)(An = 74 ='#K='E" ('5/7)

9/:%$)*

9'(G837) 6-T'RH;8*BD-D%U*&$

0n

n1)( cosn

3 (

32 )

9/:%$)*

92(G838) RH;8*BD-D%U*&$

1n 1n3

ne

1n1)(

#K='*0;BCDE" (

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36

93. RH;8* 112 + 122 + 132 + ....+ 502 $(>='#K='*0;>='E%BCDE"9=DF4%(G (

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37

96(En 39) ?7'T&0;

7/24/2019 m16 .pdf

38/38


38

98(En 40) @L6'&M'BCD>8'$*. 3 + 3.4 + 3.42+ ... + 3.419 = 420 1B. 1.2 + 2.3 + 3.4 + ... + 19. 20 = 2660

BCDE"9=DF4%(GSQ*9CD-1. *.

m16ลำดับอนุกรม.pdf

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