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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.

k=1coskk2

Short Answer

Expert verified

The series converges absolutely.

Step by step solution

01

Step 1. Given information.

Consider the given question,

k=1coskk2

02

Step 2. Consider the general series.

The general term of the series k=1ak=k=1coskk2is given below,

ak=coskk2

The limit comparison test states that for k=1ak,k=1bkbe two series with positive terms such that 0akbk for every positive integer k. If the series k=1bkconverges, then the seriesk=1akconverges.

03

Step 3. Consider the term of the given series as positive.

The given expression satisfies coskk21k2.

The series k=1bkfor the given series isrole="math" localid="1649155585096" k=1bk=k=11k2.

The series k=1bk=k=11k2 is convergent by p-series test.

The above series is convergent and converges absolutely.

Hence, the given series is absolutely convergent.

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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

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.

0.199999...

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?

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.)

Find the values of x for which the series k=0cosx2kconverges.

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