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Math Help - Convergent series

  1. #1
    Junior Member
    Joined
    Oct 2010
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    43

    Convergent series

    Show that if a_n \geq 0 and \sum_{n=1}^{\infty}a_n<\infty}, then we can find a b_n sequence that \frac{b_n}{a_n} \rightarrow \infty, but \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; February 21st 2011 at 11:49 AM.
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  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    If q>0

    <br />
\displaystyle<br />
a_n \sim \frac{1}{n^{1+q}}<br />

    then

    <br />
\displaystyle<br />
b_n \sim \frac{1}{n^{1+q/2}}<br />
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