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Math Help - convergence of an infinite series

  1. #1
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    convergence of an infinite series

    Hi, I have some difficulty proving the following

    Decide when \sum _{n=1} ^{\infty} \frac{n^{n}}{a^{n} n!} converges when the ratio test fails.

    I think that it will diverge, but I'm not sure how to prove it.
    Thanks in advance
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  2. #2
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    The ratio test fails when a=\pm e. At a=e, use the comparison test to compare the series with \sum n^{-1/2}, using Stirling's formula. At a=-e, use the alternating series test.
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