Suppose $\sum_{n=1}^{\infty}a_n$ converges, where $ a_n\geq0$ for all $n\in \Bbb{N}$. Prove that $$ \sum_{n=1}^{\infty} \frac{a_n}{1+a_n} $$ converges.

pf: I'm thinking this is a comparison test?


1 Answer 1


Hint: Since $a_n \geq 0$ for every $n$, it must be the case that \begin{equation} \frac{a_n}{1 + a_n} \leq a_n. \end{equation}

Can you take it from here?

  • $\begingroup$ a little bit more..please $\endgroup$
    – john
    Nov 13, 2014 at 3:34
  • $\begingroup$ Do you see a way to use my hint in conjunction with your idea of applying the comparison test? $\endgroup$ Nov 13, 2014 at 3:35
  • $\begingroup$ @yoknapatawpha - mad respect for your username. $\endgroup$
    – Titus
    Nov 13, 2014 at 3:37
  • $\begingroup$ @john Do you know the conditions we need to use the comparison test? I think since $\sum a_n$ converges and $\frac{a_n}{1 + a_n} \leq a_n$, we can directly apply the comparison test to this fact alone. $\endgroup$ Nov 13, 2014 at 3:49
  • $\begingroup$ $0 \leq \frac{a_n}{1+a_n} \leq a_n $ $ $ now that $a_n$ $ $ is converges then by comparison test $ $ $b_n = a_n$ so if $b_n$ is converges then $a_n$ is converges $\endgroup$
    – john
    Nov 13, 2014 at 6:44

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