Limit Comparison Question

Question

I have a interesting problem in my book.

It states:

Show that if $a_n > 0$ and $\lim\limits_{n\to\infty} (n \cdot a_n) \neq 0$, then $\sum a_n$ is divergent.

It hints at using limit comparison buy I'm not sure about how to go about it.

Answer 1

Hints:

Let $\alpha = \lim (n a_n ) $. We are given $\alpha \neq 0$. Suppose $\alpha > 0$, and pick $\epsilon = \frac{ \alpha}{2} $. Then we can choose $N$ such that for all $n > N$, then

$$ |n a_n - \alpha | < \epsilon = \frac{\alpha}{2} \iff n a_n > \alpha - \frac{\alpha}{2} = \frac{ \alpha}{2} \iff a_n > \frac{ \alpha/2}{n} $$

Since $\sum \frac{1}{n} $ is divergent, then by comparison test, $\sum a_n$ must be divergent.