# How to prove that $\lim (n a_n) = 0$? [duplicate]

Possible Duplicate:
Series converges implies $\lim{n a_n} = 0$

How do I show that the following:

If $a_1 \geq a_2\geq ... \geq a_n\geq...$ and $\sum a_n$ converges then $\lim (n a_n) = 0$?

Thanks.

## marked as duplicate by Martin Sleziak, Sasha, Davide Giraudo, Jonas Meyer, David MitraJan 14 '12 at 18:41

• Since $a_n\to 0$ (this is necessary for the convergence of series), the monotonicity implies $a_n\ge 0$. With this additional condition, this is precisely the same question as Series converges implies $\lim{n a_n} = 0$. – Martin Sleziak Jan 14 '12 at 18:23