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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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