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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
    
    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.

    © BrainMass Inc. brainmass.com February 24, 2021, 2:16 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/infinite-series-real-numbers-absolute-convergence-10443

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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

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