# Infinite Series of Real Numbers (Absolute Convergence)

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Define ak recursively by a1 = 1 and

ak = (−1)k

1 + k sin

1

k

−1

ak−1, k > 1.

Prove that

P

1k

=1 ak converges absolutely.

Since this problem is an analysis problem, please be sure to be rigorous.

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##### Solution Summary

Absolute convergence is proven for an infinie series of real numbers.

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Please see the attached file for the complete solution.

Thanks for using BrainMass.

We have to consider the series:

First of all from the given information that a1=1, we see that for k>1:

and then it turns out that:

...

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