Alternating series variant of a convergent series

Question

I'm trying to prove whether the following series is convergent, divergent, or that there is not enough info.

If the series ∑ a_n is convergent and has positive terms, what is the series below?

$$ \sum{(-1)^na_n}$$

I know that the series a_n is absolutely convergent, so the alternating one should be convergent, but I"m not sure how to prove it. The limit of a_n as n goes to infinity is 0, so a_n should be decreasing eventually, so it should be convergent by the Alternating Series Test.

Answer 1

The general rule is that if $\sum|x_n|$ converges, then $\sum x_n$ converges. This is a simple application of that fact.

Answer 2

By the convergence of $\sum_0^\infty a_k$, the sequence $(s_n)$, where $s_n=\sum_0^n (-1)^k a_k$ is a Cauchy sequence.

Proof: If $m\lt n$, then $|s_n-s_m|\le a_{m+1}+\cdots +a_n$.