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Chapter 7: Q. 57 (page 632)

In Exercises 56 and 57 we ask you to complete the proof of Theorem 7.33. For these exercises let ∑k=1∞akand ∑k=1∞bkbe two series with positive terms.

Show that if limk→∞akbk=0and ∑k=1∞bkconverges, then localid="1649095537930" ∑k=1∞akconverges.

Short Answer

Expert verified

As limk→∞akbk=0so a positive integer Nalso exists such that 0<ak<bkfor allk>N.

According to The Comparison Test, if ∑k=1∞ak&∑k=1∞bktwo series with nonnegative terms where 0<ak<bkand ∑k=1∞bkconverge, then the series also ∑k=1∞akconverges.

Step by step solution

01

Step 1. Given Information.  

The given series is ∑k=1∞ak.

series∑k=1∞bkconverses andlimk→∞akbk=0.

02

Step 2. Proof.

As given limk→∞akbk=0so a positive integer Nalso exists such that localid="1649096513563" 0<ak<bkfor all k>N.

According to The Comparison Test, if ∑k=1∞ak&∑k=1∞bktwo series with nonnegative terms where localid="1649096538086" 0<ak<bkand∑k=1∞bkconverge, then the series ∑k=1∞akalso converges.

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Most popular questions from this chapter

Determine whether the series ∑k=0∞2k+25k-1converges or diverges. Give the sum of the convergent series.

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,∞)such that limx→∞f(x)=α>0, What can the integral tells us about the series∑k=1∞f(k) ?

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

Determine whether the series ∑k=0∞4k+132kconverges 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.
See all solutions

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