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Math Help - Prove inequality?

  1. #1
    Super Member fardeen_gen's Avatar
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    Prove inequality?

    If n is a positive integer and S>1, prove that:

    \frac{1}{1^S} + \frac{1}{2^S} + \frac{1}{3^S} + \mbox{...} + \frac{1}{n^S} < \frac{1}{1 - 2^{1 - S}}
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    If n is a positive integer and S>1, prove that:

    \frac{1}{1^S} + \frac{1}{2^S} + \frac{1}{3^S} + \mbox{...} + \frac{1}{n^S} < \frac{1}{1 - 2^{1 - S}}
    Hi fardeen_gen.

    Observe that

    \frac1{3^s}\ <\ \frac1{2^s}

    \frac1{5^s}\ <\ \frac1{4^s}

    \frac1{7^s}\ <\ \frac1{6^s}

    \vdots

    \therefore\ 1-\frac1{2^2}+\frac1{3^s}-\frac1{4^s}+\cdots+\frac1{(2k)^s}\ <\ 1

    \implies\ \sum_{n\,=\,1}^\infty\frac{(-1)^{n+1}}{n^s}\ <\ 1

    \therefore\ \frac1{1^s}+\frac1{2^2}+\cdots+\frac1{n^s}\ <\ \zeta(s)\ =\ \frac1{1-2^{1-s}}\sum_{n\,=\,1}^\infty\frac{(-1)^{n+1}}{n^s}\ <\ \frac1{1-2^{1-s}}
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