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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=1akin which ak0.

(b) A divergent p-series.

(c) A convergent p-series.

Short Answer

Expert verified

(a) The example of the series is k=1ak=k=11k.

(b) The example of the series is k=1ak=k=11k.

(c) The example of the series is k=1ak=k=11k2.

Step by step solution

01

Part (a) Step 1. Given Information.

A divergent series:

k=1ak

And ak0

02

Part (a) Step 2. Consider the given series.

Consider the given series.

k=1ak=k=11k=limk1k=0

03

Part (a) Step 3. Find the series.

So by using the harmonic series and p-test series, the series k=1ak=k=11kis divergent.

04

Part (b) Step 1. Find an example.

Consider the series,

k=1ak=k=11k

which is a harmonic series, and by p-series test, the series is divergent.

05

Part (c) Step 1. Consider the series.

Consider the series,

k=1ak=k=11k2

which is convergent since p=2>1.

So the convergent p-series test is k=1ak=k=11k2.

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Most popular questions from this chapter

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=11k2

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=3hk=10.11k.

(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=6111hk=10.11k.

(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?

Explain how you could adapt the integral test to analyze a series k=1f(k)in which the functionf:[1,) is continuous, negative, and increasing.

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

k=112k3/4

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=112k1, the sum will be greater than α. Then argue that, by adding in some other finite number of the terms of

k=112k , 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.)

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