Show that if series $a_n$ converges then series $(-1)^n a_n$ converges - True or false?

Question

Show that if series $a_n$ converges then series $(-1)^n a_n$ converges.

Is that true? A quick proof is highly appreciated.Thanks

Answer 1

This is not true. Let $a_n = (-1)^n \frac{1}{n}$. As

1. $\lim_{n \rightarrow \infty }\frac{1}{n} = 0$,

2. $\frac{1}{n} >0$ for any $n \geq 1$,

and

3. $\frac{1}{n+1}<\frac{1}{n}$ for any $n \geq 1$,

we know by alternating series test that $\sum_{n=1}^\infty a_n$ converges. However

$\sum_{n=1}^\infty (-1)^n a_n = \sum_{n=1}^\infty \frac{1}{n}$, and this does not converge.

We have identified a sequence $a_n$ such that $\sum_{n=1}^\infty a_n$ converges but $\sum_{n=1}^\infty (-1)^n a_n$ diverges.

Please let me know if there is any clarification necessary.