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Math Help - a critical point(?) to a sum of series being convergent or divergent ;;

  1. #1
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    Question HELP! a critical point(?) to a sum of series converging or diverging...

    Well, I got curious about this...

    It is quite easy to show that 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to the positive infinity

    and also, the fact that 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... converges to a value pi^2/2, a.k.a basel problem is widely known.

    Then, will there be a 'critical point(?)' s.t.

    lim(n -> infinity) sigma[1/(n^a)]

    (i.e. 1/1^a + 1/2^a + 1/3^a +1/4^a + ...)

    be convergent or divergent ???

    Help! ><
    Last edited by Physicsdochi; July 23rd 2012 at 01:24 AM.
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  2. #2
    Super Member girdav's Avatar
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    Re: a critical point(?) to a sum of series being convergent or divergent ;;

    Discuss the cases a\leq 1 and a>1.
    Thanks from Physicsdochi
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  3. #3
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    Re: a critical point(?) to a sum of series being convergent or divergent ;;

    AHA! wow!!! I didn't even realize that the question is related to the zeta function! z(s)
    When a <= 1, it diverges and when a>1, it converges... now i get it
    Thanks! =)
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