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Chapter 7: Q. 59 (page 640)

Use any convergence test from Sections 7.4–7.6 to determine whether the series in Exercises 41–59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.

∑k=1∞1·4·7···(3k-2)3·6·9···3k

Short Answer

Expert verified

The given series diverges.

Step by step solution

01

Step 1. Given Information.  

The given series is∑k=1∞1·4·7···(3k-2)3·6·9···3k.

02

Step 2. Determine whether the series converges or diverges.

To determine whether the series converges or diverges we will use the comparison test since the series has positive terms that meet the hypothesis of the test.

Let ∑k=1∞ak=∑k=1∞3k+13k+3and∑k=1∞bk=∑k=1∞13k.

So, 0≤3k+13k+3≤13k.

We can write

∑k=1∞bk=∑k=1∞13kas∑k=1∞bk=13∑k=1∞1k.

Now, ∑k=1∞bk=∑k=1∞1kisoftheform∑k=1∞bk=∑k=1∞1k.

If p = 1 then the series diverges.

Here p=1,thus, ∑k=1∞bk=∑k=1∞13kdiverges.

By the comparison test, ∑k=1∞ak=∑k=1∞3k+13k+3also diverges.

Thus, the given series diverges.

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Most popular questions from this chapter

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.

1.272727...

Prove Theorem 7.31. That is, show that if a function a is continuous, positive, and decreasing, and if the improper integral ∫1∞a(x)dxconverges, then the nth remainder, Rn, for the series∑k=1∞a(k) is bounded by0≤Rn=∑k=n+1∞a(k)≤∫n∞a(x)dx

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

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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.

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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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