# Infinite Series of Real Numbers (Absolute Convergence)

Please see the attached file for the fully formatted problem.

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.

https://brainmass.com/math/real-analysis/infinite-series-real-numbers-absolute-convergence-10443

#### Solution Preview

Please see the attached file for the complete solution.

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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:

...

#### Solution Summary

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

$2.19