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Chapter 7: Q. 71 (page 654)

Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series

Short Answer

Expert verified

Hence, the series is divergent.

Step by step solution

01

Step 1. Given information

Changing the order of the summands in a conditionally convergent series can change the value of the sum.

02

Step 2. Proof

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Most popular questions from this chapter

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak鈫�0, then 鈭�k=1鈭�akconverges.

(b) True or False: If 鈭�k=1鈭�akconverges, then ak鈫�0.

(c) True or False: The improper integral 鈭�1鈭�f(x)dxconverges if and only if the series 鈭�k=1鈭�f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series 鈭�k=1鈭�k-pconverges.

(f) True or False: If f(x)鈫�0as x鈫�鈭�, then 鈭�k=1鈭�f(k) converges.

(g) True or False: If 鈭�k=1鈭�f(k)converges, then f(x)鈫�0as x鈫�鈭�.

(h) True or False: If 鈭�k=1鈭�ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

Explain how you could adapt the integral test to analyze a series 鈭�k=1鈭�f(k)in which the functionf:[1,鈭�)鈫�鈩� is continuous, negative, and increasing.

Whenever a certain ball is dropped, it always rebounds to a height60% of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of 1 meter?

Let 鈭�k=1鈭�akbeaconvergentseriesand鈭�k=1鈭�bkbeadivergentseries.Prove that the series 鈭�k=1鈭�ak+bkdiverges.

Let 伪be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to 伪. (Hint: Argue that if you add up some finite number of the terms of 鈭�k=1鈭�鈥�12k鈭�1, the sum will be greater than 伪. Then argue that, by adding in some other finite number of the terms of

鈭�k=1鈭�鈥�鈭�12k , you can get the sum to be less than 伪. By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to 伪.)

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