Leibniz criterion for alternating series says that the series $$ \sum_{n=0}^{+\infty}(-1)^n a_n $$ conditionally converges if the following two hypothesis are verified:
- $\lim_{n\to+\infty}a_n =0$,
- $a_n$ is monotonically decreasing.
My question is: if $b_n$ is asymptotic to $a_n$ (which means that $\lim_{n\to+\infty}a_n/b_n=1$), can I study the monotonicity of $b_n$ instead of $a_n$?
I know the proof of the Leibniz criterion, and I am aware of the fact that if $b_n$ is asymptotic to $a_n$ this doesn't mean that $b_n$ is monotonically decreasing.
What I'm searching for is a counterexample: I would like to find a non converging alternating series $\sum_{n=0}^{+\infty}(-1)^na_n$ and a (conditionally) convergent alternating series $\sum_{n=0}^{+\infty}(-1)^nb_n$ where $b_n$ is asymptotic to $a_n$.