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91Ó°ÊÓ

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31–48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

∑k=1∞cos1k

Short Answer

Expert verified

The series ∑k=1∞cos1k is Divergent.

Step by step solution

01

Step 1. Given information  

We are given,

∑k=1∞cos1k

02

Step 2. Checking the Convergence and Divergence 

The terms of the series ∑k=1∞cos1kare positive.

The expression cos1ksatisfles the following inequality,

cos1k≤1k

The series ∑k=1∞bkfor the series ∑k=1∞cos1kis given by:

∑k=1∞bk=∑k=1∞1k

The series ∑k=1∞bk=∑k=1∞1k is divergent by p-series test.

Therefore, the series ∑k=1∞akis also divergent.

Hence, the series ∑k=1∞cos1kis divergent

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