Lesson 14 & Lesson 15

8/3/2019 Lesson 14 & Lesson 15

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14.6. SUGGESTED PROBLEMS 169

The famous German mathematician Bernhard Riemann (1826-1866) proved that

the terms of a conditionally convergent series can be rearranged so that the re-arranged series has any prescribed sum 8, including 00 and -00. A conditionallyconvergent series can also be rearranged so that the resulting series does not

have a sum.

14.6 Suggested Problems 2_l1; 7- 9;/lJo,//9b1. Give brief clear answers to the following questions:

(a) What is an alternating series?(b) To which alternating series does the alternating series test apply?

(c) Suppose the hypotheses of the alternating series test are met. Whatcan you say about the error when the alternating sum of the first n

terms of the series are used to approximate the sum of the series?Show that each\alter~atiIl~,series .converges. Then :stimate the errormade when the Ì)tial sum 810 is used to approximate the sum ofthe series.

2 ,"00 ( )n+l 1. L.m=1 -1 In(n+1)

.i 3. L:=1 (-It+1 n2~1

4. L:=1 (-lr

5 ,"00 (_I)n+ln. L.n= 1 2n

6 ,"00 (-wL.n=O vn( n+1)

(a) Determine whether the series is absolutely convergent, conditionallyconvergent, or divergent.

,. - -A ,"00 (_I)n-l-l\ì - VI L.n=1 nvn

~. L:=1 (-It+1 5¡n

i L:=1 (-It+1 5~n

10 ,"00 ~L.n= 1 n

11 ,"00 sin(mr/2)

. L.n=l n12 ,"00 (_I)n+l sin(n)L.n=l nvn

13 ,"00 (_I)n+l 1. L.n=l vn(n+l)



170 LESSON 14. ALTERNATING SERIES, ABSOLUTE CONVERGENCE

14 ,"00 (_2)n-lL.n=l n2

(a) Determine the values of x for which the series converges. (b) Determinewhere the convergence is absolute. (c) Draw a figure that ilustrates thedomain of convergence.

15. L:=l xn In2

16. L:=l xn In

17. L:=l (sin nx) jn3

18. L:=l xn l..

(a) Verify that the series converges by the alternating series test. (b) Findthe smallest N such that the error estimate 18 - 8N I -( 0.001 is guaranteedto hold in connection with the alternating series test. (c) Calculate thecorresponding 8N correct to five decimal places. (d) Is 8N :; 8 or is

8N -( 8? Explain briefly.

19. L:=l (-It+1 In3

20. L:=l (_I)n+l Inn

21. L:=l (-It+1 (sin (1/n)) In2

Let z be a complex variable. Determine where each series is absolutelyconvergent. Describe the domain of absolute convergence geometricaly.

22. L:=l zn In2

23. L:=l zn In

24. L:'=l zn l..

25. L:'=o (-it znln..

26. Is the 75th partial sum 875 ofthe series L:=l (-It+1 I (n2 + 1) an over-estimate or an underestimate for the sum 8 of the series? Explain.

27. Is the 56th partial sum 856 ofthe series L:=l (-It+1 I (n2 + 1) an over-estimate or an underestimate for the sum 8 of the series? Explain.

~~"'(' ¡i;. -_,

:;~~. ?~



178 LESSON 15. ROOT AND RATIO TESTS

15.6 Special Limits

Here are several special

limits that you may find useful when you apply the ratioor root test to particular series. (If you did the group project on special

limitsin the supplemental materials at the end of these lessons, you already have metmost of the special limits and developed some numerical evidence in support ofthem. Now you can prove some of them!)

As n -4 00 :

1. an -4 0 provided lal -( 1.

2. a1jn -. 1 provided a:; O.

for any p:; 0; in particular, In n -4 O.n

3. In n -4 0nP

4. n1jn -4 1.

5. (1 + ~)n -. eX for all x.

xn6. -i -. 0 for all x.

n.

A homework problem guides you through the verification of three of theseresults.

15.7 Suggested Problems

What can you say about the series L an in each case.

1. liID-+oo I a~~l I = 4

2. limn-+oo I a~~ 1 I = l

3. limn-+oo I a~~i I = 1

Use either the ratio or the root test to determine (absolute) convergence,divergence, or that the test is inconclusive.

4. Ln (lr

5. L (In1nr

6. L(2n~llr

7. L(-lt G + ~r



15.7. SUGGESTED PROBLEMS 179

8. L ~~

9. L(-lt (;~)!

10. L (;~)!

'" nn11. L. n!

12. 1 + ~ + l + i + . . .

13. 1 - i2 + l2 - 4\ + . . .

14. 1- 2~ + 3~ - 4À +...

15 -- _ -- + ~ _ -- + _ . . .1.3,4 2.4.5 3,5.6 4,6.7

16. 1.L + 2.~.5 + 3,:,6 + 4.~.7 + . . .

Determine whether the series is absolutely convergent, conditionally con-vergent, or divergent.

00 (_2)n17. Ln=l n2

00 (_2)n18. Ln=O n!

19. L:=l (2~2~~) n

20. L:=l (-it (ý~:~~l) n

(a) Use the ratio test to show that the series converges. (b) Then use theerror estimate in this lesson associated with the ratio test to approximatethe sum to within 0.001.

)

/'__ 00';~::1. Ln=O Ijn!

22. L:=o nj2n

23. L:=o (n2 + n) j3n

24. L:=o (-it (l¡n r

(a) Use the root test to show that the series converges. (b) Then use the

error estimate in this lesson associated with the root test to approximatethe sum to within 0.001.

25. L;:o Ijn! Hint. \f is increasing.

26. L:=onj2n



180LESSON 15. ROOT AND RATIO TESTS

27. L:=o (_i)n (l¡n)n

28. L:=o (n2 + n) j3n

29. For real x, L:=l xnjn3n is a real power series. (a) Show that it converges

absolutely by the ratio test for all ixl -( 3. (b) Its Sum S (x) and partialsums S N (x) are functions of x. Determine N so that that the partial

sum SN(X) wil approximate S (x) to within 0.001 for all x in the interval-2:: x:: 2.

30. For real x, L~l nxn j3n is a real power series. (a) Show that it converges

absolutely by the root test for all ¡xl -( 3. (b) Its sum S (x) and partialsums SN (x) are functions of x. Determine N so that that the partial

sum SN(X) wil approximate S (x) to within 0.001 for all x in the interval-2:: x:: 2.

31. Use the ratio test to determine where the given power series converges

absolutely and where it diverges. What happens at the two endpoints ofthe interval of convergence?

(a) Lnxn

(b) Ln2xn

(c) Ln3xn

(d) What general result is suggested by (a), (b), and (c)? State andconfirm it.

32. Use the root test to determine where the given power series converges

absolutely and where it diverges. What happens at the two endpoints ofthe interval of convergence?

(a) Lxnjn

(b) Lxnjn2

(c) Lxnjn3

(d) What general result is suggested by (a), (b), and (c)? State andconfirm it.

33. Fix ¡xl -( 1 and p :; O. Use Problem 31( d) to show that nPxn -4 0 asn -4 00.

(a) Determine for which z (as usual a complex vaiable) the series Con-

verges absolutely and for which z it diverges. (b) Describe the region ofabsolute convergence geometrically.

34. L(-I)nn2zn/2n



15.7. SUGGESTED PROBLEMS181

35. L (In n) zn

36. Lznjn!37. L n!zn j3n

38. The terms of the series L an are defined recursively by ai = Ij2 and

an+1 = j~~~an for n ~ 1. Determine whether the series converges ordiverges.

39. The terms of the series L an are defined recursively by ai = 2 and an+! =

~~~i an for n ~ 1. Determine whether the series converges or diverges.

40. (More Error Estimates) Assume there is a positive integer N such thatan ~ an+i :; 0 for all n :; N. Establish the following assertions:

aN+irN+1(a) If 0, r , 1, then L:=N+1 anrn $ l-r

rn rN+1(b) Ifr:; 0; then L:=N+1anl ,aN+1(N )leT.n. + 1.

" 00 rn rN+i N + 2c) Moreover, in (b) ifr , N+2, then Ln=N+l an n! ,aN+l (N + I)! N + 2 _ r'

Hint. Use a geometric series.

41. Let SN (x) be the Nth partial sum and S (x) the sum of a given power

series. Do the following:

(a) For the power series L:=l 2": xn use (a) of the previous problem to

find a reasonable N so that SN (x) approximates S (x) to 4 decimalplaces (error at most 0.00005) for al x with - 1 $ x $ 1. Hint. For

ixl $ 1,

In nl n n (l)nnx $ 2n = (J2t J2

(b) For the power series L:=l In ~~ use (b) of the previous problem to

find a reasonable N so that S N (x) approxiates S (x) to 4 decimalplaces (error at most 0.00005) for all x with -2 $ x $ 2.

(c) For the power series L:=l In ~~ use (c) of the previous problem to

fid a reasonable N so that SN (x) approximates S (x) to 4 decimal

places (error at most 0.00005) for all x with -2 $ x $ 2.

42. Use Stirling's formula to deduce that y' -. 00 as n -4 00.

43. (The Special Limits) Recall that eX -.0 as x -4 -00, eX -4 1 as x -4 0,

and aX = ex1na for any a? O.

(a) Verify special limit #1. Hint. lanl = lain = enlnlal

Lesson 14 & Lesson 15

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