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Thread: Convergent series

  1. #1
    Junior Member
    Joined
    Oct 2010
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    43

    Convergent series

    Show that if $\displaystyle a_n \geq 0$ and $\displaystyle \sum_{n=1}^{\infty}a_n<\infty}$, then we can find a $\displaystyle b_n$ sequence that $\displaystyle \frac{b_n}{a_n} \rightarrow \infty$, but $\displaystyle \sum_{n=1}^{\infty}b_n<\infty$ is true.
    I.e for any convergent series, exists an asymptotically greater convergent series.

    Thank you in advance!
    Last edited by zadir; Feb 21st 2011 at 11:49 AM.
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  2. #2
    Senior Member
    Joined
    Mar 2010
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    If q>0

    $\displaystyle
    \displaystyle
    a_n \sim \frac{1}{n^{1+q}}
    $

    then

    $\displaystyle
    \displaystyle
    b_n \sim \frac{1}{n^{1+q/2}}
    $
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