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Math Help - The series converges?

  1. #1
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    The series converges?

    Let a_n be positive, and \sum_{n=1}^\infty a_n be divergent. Assume S_n=a_1+\cdots+a_n. Prove that for any \sigma>1, the series \sum_{n=1}^\infty \frac{a_n}{S_n^\sigma} converges.
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  2. #2
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    Re: The series converges?

    Hey xinglongdada.

    Consider that a/n + b/n = (a+b)/n and then consider the ratio test.
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  3. #3
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    Re: The series converges?

    Do you mean by considering \sum_{n=1}^\infty (S_n-S_{n-1})/S_n^\sigma, then using ratio criterion? I do not know how to do. Would you be more precise?
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  4. #4
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    Re: The series converges?

    Basically the ratio criterion is that if |a_(n+1)/a_n| < 1 then the series converges where a_n is the n_th term of the series.

    So basically you need to show that this happens given your series expansion.
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  5. #5
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    Re: The series converges?

    In fact, we need have |a_{n+1}/a_n|<r<1 for some fixed r to guarantee the convergence of the series. So I do not know how to take the limit...
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