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Chapter 7: Q. 45 (page 632)

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31–48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

∑k=1∞1kek

Short Answer

Expert verified

The series ∑k=1∞1kekis Convergent.

Step by step solution

01

Step 1. Given information  

We are given,

∑k=1∞1kek

02

Step 2. Checking the Convergence and Divergence 

Evaluating the integral, integrate by parts.

So,

∫x=1∞f(x)dx=limk→∞∫x=1k1xexdx

Putting x=u⇒12xdx=du,

role="math" localid="1649317672240" =2limk→∞∫u=1ke-udu=2limk→∞-e-n1k=2limk→∞-e-k+e

Taking limit,

=2e

03

Step 3. Checking the Convergence and Divergence 

Thus, the value of the integral is ∫x=1∞1xexdx=2e.

The integral converges. Therefore, the series ∑k=1∞1kekis convergent. Hence, by integral test, the series ∑k=1∞1kekis convergent.

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Most popular questions from this chapter

Determine whether the series ∑k=0∞-3k+14k-2converges or diverges. Give the sum of the convergent series.

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.

Which p-series converge and which diverge?

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.

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

0.6345345...
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