If the series $\sum n a_n$ is convergent, then what can we say about $\sum a_n$ ? I think $\sum a_n$ must converge to zero. Here is my proof.
$\sum n a_n$ is convergent
$\Rightarrow$ for $\epsilon \gt 0$ $ \exists$ $ N \in \mathbb N$ such that
$l-\epsilon \lt n a_n \lt l+\epsilon$ for all $n \geq N $
$\Rightarrow \frac{l-\epsilon}{n} \lt a_n\lt\frac{l+\epsilon}{n} $ for all $n \geq N$
Now, $\epsilon$ and $l$ are fixed. So, by sandwich theorem for sequences, we can conclude that $a_n$ converges to zero.
Am I correct? Thanks.
There may be similar question like this. But I am asking whether this proof is correct or not. So, this is not duplicate.
Best Answer
Let consider
$$a_n=\frac1{n^3}\implies \sum n a_n =\sum \frac1{n^2} $$
which converges but
$$\sum \frac1{n^3}$$
doesn't converges to zero.