Example for this criterion?

Question

Here is a powerful criterion on sequences and series :

Let $(a_n)_{n\in \mathbb{N}} \in \mathbb{R^+}^{\mathbb{N}}$ a decreasing sequence. If $A_n = \sum \limits_{k=0}^{n}a_k$ converges then $a_n = o(\frac{1}{n})$.

I was looking for an interesting example which uses this criterion.

Thanks in advance !

Answer 1

Interestingly, while the harmonic series $\sum\frac1n$ diverges (and, of course, $\frac1n\not=\mathcal o (\frac1n)$), $\sum(\frac1n)^{1+\epsilon}$ converges for any $\epsilon \gt0$.

So the condition you have given is in a sense optimal...