A Series Fails The Test For Divergence, but is Still Divergent?

Question

I know $ \sum_{n=2}^\infty \frac{1}{n*ln(n)} $ is divergent by the integral test or comparison test; however, I notice that it fails the Series Test For Divergence ($\lim_{n\to\infty}a_n \neq 0 \Rightarrow Divergence$). Can a series fail this test and still diverge?

Answer 1

Yes, decay of the summand as $ n \rightarrow \infty$ is a necessary, not sufficient condition for convergence of the series.

Answer 2

It goes in only one way. If $lim_{n\to \infty} a_n \neq 0$ then the series diverge. It's a minimal criterial, not a sufficient condition

Answer 3

a simple example is summation of $\frac{1}{n}$