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Chapter 7: Q. 9 (page 631)

If you suspect that a series ∑k=1∞ak converges, explain why you would want to compare the series with a convergent series, using either the comparison test or the limit comparison test.

Short Answer

Expert verified

If the series ∑k=1∞akis convergent, comparing it to divergent series will yield no results since the behaviour of the series ∑k=1∞akis dependent on the behaviour of the series ∑k=1∞bk.

As a result, if the ∑k=1∞akseries converges, it must be compared to a convergent series.

Step by step solution

01

Step 1. Given information

A series is given as∑k=1∞ak

02

Step 2. Verification

The limit comparison test for ∑k=1∞akand ∑k=1∞bkare the series having positive terms the the following conditions may apply,

If limk→∞akbk=L, L must be positive number then it may be either converging or diverging.

If limk→∞akbk=0then if ∑k=1∞bkconverges ∑k=1∞akconverges

If limk→∞akbk=∞then if ∑k=1∞bkdiverges ∑k=1∞akdiverges

If the series ∑k=1∞akis convergent, comparing it to divergent series will yield no results since the behaviour of the series ∑k=1∞akis dependent on the behaviour of the series ∑k=1∞bk.

As a result, if the ∑k=1∞akseries converges, it must be compared to a convergent series.

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Most popular questions from this chapter

Prove that if ∑k=1∞akconverges to L and ∑k=1∞bkconverges to M , then the series∑k=1∞ak+bk=L+M.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series ∑k=1∞akin which ak→0.

(b) A divergent p-series.

(c) A convergent p-series.

Let αbe any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to α. (Hint: Argue that if you add up some finite number of the terms of ∑k=1∞ 12k−1, the sum will be greater than α. Then argue that, by adding in some other finite number of the terms of

∑k=1∞ −12k , you can get the sum to be less than α. By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to α.)

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?

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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