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If $a_n>0$ for all n, and $\sum_{n=1}^\infty a_n$ diverges, then can I prove $\sum_ {n=1}^\infty \frac{a_n}{a_n+1}$ diverges by showing that $\frac{a_n}{a_n+1}$ = $1-\frac{1}{a_n+1}$ and then use the comparison test?

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    In both cases, $a_n$ does not bound $\frac{a_n}{a_n+1}$ from above, right? So how can I use the comparison test in this case?2012-11-14

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First prove that $\frac{a_n}{a_n+1}\xrightarrow[n\to\infty]{} 0\iff a_n\xrightarrow[n\to\infty]{}0$ and then use limit comparison test to show that

$\sum_{n=1}^\infty a_n$ converges iff $\sum_ {n=1}^\infty \frac{a_n}{a_n+1}$ converges:$\displaystyle{\lim_{n\to \infty}\dfrac{\frac{a_n}{1}}{\frac{a_n}{a_n+1}}}=\lim_{n\to \infty}a_n+1=1 \neq 0.$

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    @Alti: You are welcome.2012-11-14