Is it true? If $\lim_{n\to\infty}\frac{|a_{n}|}{|c_{n}|}=0$, then the series $\sum_{n=1}^\infty a_n$ converges

Question

Let $\{c_n\}$ be a sequence that converges to $0$ and let $\{a_n\}$ be a sequence for which the following applies: $\lim_{n\to\infty}\frac{|a_{n}|}{|c_{n}|}=0$. Then the series $\sum_{n=1}^\infty a_n$ converges.

I tried to think of a counter-example to show that this is false but I can't really come up with any. I think it's false, because the only thing that this statement tells me is that the sequence $\{a_n\}$ converges faster to $0$ than $\{c_n\}$. And this doesn't really tell me anything about the convergence of the series.

Any help would be appreciated, thanks!

Answer 1

Note that in case $\sum c_n$ is absolutely convergent, we can conclude with the argument below:

$\lim_{n\to\infty}\frac{|a_{n}|}{|c_{n}|}=0$ means that for $n_0$ large enough we have $\forall n\ge n_0, \dfrac{|a_n|}{|c_n|}<1$.

In particular $\sum\limits_{n_0}^{\infty}|a_n|<\sum\limits_{n_0}^{\infty}|c_n|<+\infty$ thus $\sum a_n$ is absolutely convergent too.

In case of semi-convergence of the series you can take $c_n=\dfrac{(-1)^n}{\sqrt{n}}$ and $a_n=\dfrac 1n$ as a counterexample.

And of course if you do not even suppose that $\sum c_n$ converge, there is really no reason it should work in general.

For instance take series like $\sum\frac 1n$, $\sum\frac 1{n\ln(n)}$, $\sum \frac 1{n\ln(n)\ln(\ln(n))}$, $\cdots$ none is convergent despite terms being little-o of the previous one.