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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.

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Solution Summary

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

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