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Chapter 7: Q. 14 (page 614)

Let α ∈ R. Explain why you can find a series∑k=1∞ ak

with all nonzero terms that converges to α. You may wish

to use your answer to Exercise 13.

Short Answer

Expert verified

The geometric series∑k=1∞ ak with all non-zero terms that converges toα.

Step by step solution

01

Step 1. Given information 

We have been given a series that is converges toα.

02

Step 2. Proving the series ∑k=1∞ ak. to be convergent .

Considering the series ∑k=1∞ ak.

The objective is to explain why the series converges to zero .

the series ∑k=1∞ 12kis a geometric series with constant ratio of 12which is less than 1

hence the series is convergent .

03

Step 3.Finding the sum of series 

The series ∑k=1∞ ak=∑k=1∞ 12kconverges to the sum

=1212=1

The series converges ∑k=1∞ ak=∑k=1∞ 12kthe sum of 1 .

04

Step 4. Finding the explanation of problem 

The series ∑k=1∞ ak=∑k=1∞ 12kconverges to 1.

If each term of the series is multiplied by constant the series become ∑k=1∞ αak=∑k=1∞ α2k

The convergence of the series changes from 1to α

The geometric series∑k=1∞ ak with all non-zero terms that converges toα.

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Most popular questions from this chapter

Determine whether the series ∑k=0∞4k+132kconverges or diverges. Give the sum of the convergent series.

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?

In Exercises 48–51 find all values of p so that the series converges.

∑k=1∞lnkkp

Find the values of x for which the series ∑K=0∞sinxkconverges.

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?
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