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Chapter 7: Q. 31 (page 657)

Check the convergence∑k=1∞(-1)klnk

Short Answer

Expert verified

The series diverges.

Step by step solution

01

Step 1. Given

The given series∑k=1∞(-1)klnk

02

Step 2. Checking of convergence

The general term of the series ∑k=1∞(-1)klnkis ak=lnk

localid="1649739242985" Thevalueoflimk→∞akis:limk→∞ak=limk→∞lnk=∞Sincelimk→∞ak∅0Hencetheserieslimk→∞lnkdiverges.

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

∑k=3∞ 1(k−2)2

Given a series ∑k=1∞ak, in general the divergence test is inconclusive when ak→0. For a geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

∑k=1∞ (3k−1)−2/3

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,∞)such that role="math" localid="1649081384626" limx→∞f(x)=α>0. What can the divergence test tell us about the series ∑k=1∞f(k)?

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→0, then ∑k=1∞akconverges.

(b) True or False: If ∑k=1∞akconverges, then ak→0.

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

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

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

(f) True or False: If f(x)→0as x→∞, then ∑k=1∞f(k) converges.

(g) True or False: If ∑k=1∞f(k)converges, then f(x)→0as x→∞.

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