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Chapter 7: Q 51. (page 615)

Determine whether the series ∑k=0∞πe3kconverges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series ∑k=0∞πe3k converges to 3π3-e.

Step by step solution

01

Step 1. Given information.

Given a series ∑k=0∞πe3k.

02

Step 2. Find if the series converges or not.

The series ∑k=0∞πe3kis in the standard form ∑k=0∞crkfor a geometric series with c=πand r=e3.

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

Note that 2<e<3. It follows that e3<1.

Therefore, the series localid="1648958893904" ∑k=0∞πe3kconverges.

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 ∑k=0∞πe3kconverges to π1-e3, that is 3π3-e.

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Most popular questions from this chapter

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish qkreturning each year as qk+1=(0.14(−1)k+0.36)(qk+h), where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately qe=3h∑k=1∞0.11k.

(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately qo=6111h∑k=1∞0.11k.

(c) How should Leila choose h, the number of hatchery fish to breed in order to hold the minimum number of fish returning in each run near some constant P?

For a convergent series satisfying the conditions of the integral test, why is every remainder Rnpositive? How can Rnbe used along with the term Sn from the sequence of partial sums to understand the quality of the approximation Sn?

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.

Let 0 < p < 1. Evaluate the limitlimk→∞1/klnk1/kp

Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the series∑k=2∞1klogk

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...
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