# Converges series test

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

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?

• a little bit more..please – john Nov 13 '14 at 3:34
• Do you see a way to use my hint in conjunction with your idea of applying the comparison test? – yoknapatawpha Nov 13 '14 at 3:35
• @yoknapatawpha - mad respect for your username. – Titus Nov 13 '14 at 3:37
• @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. – yoknapatawpha Nov 13 '14 at 3:49
• $0 \leq \frac{a_n}{1+a_n} \leq a_n$  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 – john Nov 13 '14 at 6:44