An exercise about convergence of series Possible Duplicate:
  Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $\sum a_n$ diverges.  




Let $a_n$ be a non-negative sequence such that the series $\sum a_n$ does not converge. Could the series $\sum a_n/(1+na_n)$ be convergent?
Hint: Try $a_n=1$ if $n$ is a power of $2$ and $0$ otherwise.
If $\sum a_n = \infty$, then there exist $n_0: n\ge n_0 \Rightarrow a_n \ge 1$. Hence,
\begin{equation}
\infty = \sum_{n\ge n_0} \dfrac{a_n}{a_n(n+1)} \le \sum_{ n \ge n_0}\dfrac{a_n}{(na_n+1)}.
\end{equation}
Then $\sum a_n/(1 + na_n)$ does not converge.

I messed up. My argument is if $\lim a_n=\infty$.