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Math Help - Mathematical Gazette slowly convergent series question

  1. #1
    Grand Panjandrum
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    Mathematical Gazette slowly convergent series question

    In the current issue of the Mathematical Gazette there is an article on slowly convergent series. There the tortoise challenges Achilles to sum the series:

     K=\sum_{n=0}^{\infty} (-1)^n (n+1)^{-\frac{n+2}{n+1}}

    to four significant figures.

    Now far as I can see; this is an alternating series with terms of strictly decreasing absolute value which go to zero as n goes to infinity and so it converges to a limit between 1 and 1-2^{-3/2} (between 1 and 0.64).

    Also numerical experiments reassuringly suggest that the limit is \approx 0.78

    However the paper seems to think the sum K \approx 1.32... Which leaves me puzzled, what have I done wrong, or what could be the misprint in the expression for the series?

    CB
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  2. #2
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    Opalg's Avatar
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    Re: Mathematical Gazette slowly convergent series question

    I agree that the limit must be between 0.64 and 1. I can't figure out where to get 1.32 from.
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