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    Infinite Series of Real Numbers (Absolute Convergence)

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    Define ak recursively by a1 = 1 and
    ak = (−1)k
    1 + k sin
    ak−1, k > 1.
    Prove that
    =1 ak converges absolutely.
    Since this problem is an analysis problem, please be sure to be rigorous.

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    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.