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Suppose $a_n > 0$ and $\sum a_n$ converges. Put $r_n = \sum_{m=n}^{\infty} a_m$.

1) Show that $\frac{a_m}{r_m} + ... + \frac{a_n}{r_n} > 1 - \frac{r_n}{r_m}$, if $m < n$, and deduce that $\sum \frac{a_n}{r_n}$ diverges.

2) Show that $\frac{a_n}{\sqrt[]r_n} < 2(\sqrt[]{r_n} - \sqrt[]{r_{n+1}})$ and deduce that $\sum \frac{a_n}{\sqrt[]r_n}$ converges.

I am stumped on this problem, do not know how to start. Any help would be great.

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    Presumably the very last summation is missing a square root symbol?2012-07-06
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    Oops..thanks Gerry!2012-07-06
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    I think you do not want the square root in the sum appearing in 1).2012-07-06
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    @David..thanks I fixed it/2012-07-06

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