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Thread: Convergent Sequences, Harmonic Series

  1. #1
    Sep 2010

    Convergent Sequences, Harmonic Series

    Prove or give a counterexample: Let an be a sequence such that the sequence bn=an+1 - an converges to 0. Does an have to converge?

    Now I know this is not true and I know that it has something to do with partial sums of the harmonic series, but it's been years since I've worked with this sort of material and I am completely lost on how to use the harmonic series to disprove the theorem.
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  2. #2
    Member Traveller's Avatar
    Sep 2010
    The solution is in fact to consider the partial sums of the harmonic series as the terms of the sequence.

    Let $\displaystyle a_n = H_n$. Since the harmonic series diverges, a_n diverges.
    Clearly, $\displaystyle b_n = a_{n+1} - a_n = \frac{1}{n+1}$ and hence converges to 0.
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