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Chapter 7: Q 48. (page 615)

Determine whether the series ∑n=1∞-1n+13nconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series ∑n=1∞-1n+13n converges to 14.

Step by step solution

01

Step 1. Given information.

Given a series ∑n=1∞-1n+13n.

02

Step 2. Find if the series converges or not.

The index starts with 1, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of geometric series is ∑k=0∞crk.

Here, the series ∑n=1∞-1n+13nhas c=13and r=-13.

The geometric series converges if and only if r<1.

Since r=-13, it follows that the series localid="1648883874265" ∑n=1∞-1n+13nconverges.

03

Step 3. Find the value to which the series converges.

If the geometric series ∑k=0∞crkconverges, it converges to c1-r.

So, the series ∑n=1∞-1n+13nconverges to 131--13, that is 14.

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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→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.

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.

∑k=1∞ k2+1k!

An Improper Integral and Infinite Series: Sketch the function f(x)=1xfor x ≥ 1 together with the graph of the terms of the series ∑k=1∞1k.Argue that for every term Snof the sequence of partial sums for this series,Sn>∫1n+11xdx. What does this result tell you about the convergence of the series?

Iflimk→∞akbk=∞and∑k=1∞akconverges, explain why we cannot draw any conclusions about the behavior of∑k=1∞bk.

Determine whether the series ∑n=4∞411nconverges or diverges. Give the sum of the convergent series.
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