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Chapter 7: Q. 4 (page 639)

Use Exercise 3 to explain why the ratio test will be inconclusive for every series ∑k=1∞akin which akis a polynomial.

Short Answer

Expert verified

If akis a polynomial then localid="1649104353595" limx→∞p(x+1)p(x)will be 1and the ratio test is inconclusive for series whenlimx→∞p(x+1)p(x)=1.

Step by step solution

01

Step 1. Given information. 

If p(x)is a polynomial then role="math" localid="1649104261634" limx→∞p(x+1)p(x)=1.

Ifakis a polynomial then the ratio test will be inconclusive for every series role="math" localid="1649104105405" ∑k=1∞ak.

02

Step 2. Verification.

Consider ak=p(x)

role="math" localid="1649104033456" ak+1=p(k+1)limk→∞ak+1ak=limk→∞p(k+1)p(k)

as limk→∞p(k+1)p(k)=1

so role="math" localid="1649104148331" limk→∞ak+1ak=1L=1

According to the ratio test, if ∑k=1∞akis a series with positive terms and L=limk→∞ak+1ak=1then then the ratio test will be inconclusive for series.

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Most popular questions from this chapter

For each series in Exercises 44–47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

∑k=2∞1k(lnk)2

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?

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.

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.

0.305130513051...

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∞1k
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