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Thread: the next part of the norm question..

  1. #1
    MHF Contributor
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    the next part of the norm question..

    http://i37.tinypic.com/2nhosqa.jpg

    i tried:
    $\displaystyle
    x_i=\frac{1}{i^s}
    $
    i from 1 to infinty
    the whole sum is called u
    $\displaystyle
    \left \| u \right \|_p=(\sum_{n=1}^{\infty}(\frac{1}{i^s})^p)^\frac{ 1}{p}$
    and i think we need to show that i converges

    but thats only a shot in the dark

    i dont have a clue about it
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  2. #2
    Moo
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    Yes that's what you had to do.

    Now, recall that $\displaystyle (a^b)^c=a^{bc}$ to get $\displaystyle \left(\frac{1}{i^s}\right)^p=\frac{1}{i^{sp}}$

    For what s (depending on p) does $\displaystyle \sum_{i=1}^\infty \frac{1}{i^{sp}}$ converge ?

    It's just a Riemann sum...

    And taking something to the power 1/p doesn't change anything in the convergence or divergence.
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