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# The series converges

Prove that the series

converges. Examine if the convergence is absolute.

Solution

Lemma: Let such that then the sequence

is bounded.

With the above lemma in mind the series converges as a consequence of Dirichlet’s theorem. The fact that the series does not converge absolutely can be seen by applying the Jordan inequality

(1)

that

hence the series diverges absolutely.

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