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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∞sin1k2

Short Answer

Expert verified

The series ∑k=1∞sin1k2is Convergent.

Step by step solution

01

Step 1. Given information  

We are given,

∑k=1∞sin1k2

02

Step 2. Checking the Convergence and Divergence  

The terms of the series ∑k=1∞sin1k2are positive.

The expression sin1k2satisfies the following inequality,

sin1k2≤1k2

The series ∑k=1∞bkfor the series ∑k=1∞sin1k2is given by,

∑k=1∞bk=∑k=1∞1k2

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

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

Hence, the series ∑k=1∞sin1k2is convergent .

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