How to give a example that $\lim\limits_{n\to\infty}n|a_{n}-a_{n+1}|=0$ and {$a_{n}$} divergent

Question

I have tried to link with the series {$\sum \frac{1}{n}$} but it does not work.

Answer 1

If we define $b_n = a_{n+1} - a_n$, then we can reframe the problem as follows: we're looking for a series $\sum_{n=1}^\infty b_n$ that fails to converge but satisfies $\lim_{n \to \infty}\frac{b_n}{1/n} = 0$.

In other words, we need a series for which $b_n$ is "smaller" than $1/n$, but still diverges. One working example is $b_n = \frac{1}{n\log n}$.