Divergence of alternating series

Question

Say I have an alternating series $\sum (-1)^n a_n$. If $\lim_{n \rightarrow \infty} a_n=\infty$ or if it does not tend to $0$, then, does $\sum (-1)^n a_n$ diverge?? I know that if this holds then $\sum a_n$ diverges, but if I add this $(-1)^n$ would that still be divergent?

Answer 1

Yes, it still diverges, because you don't have $\lim_{n\to\infty}\bigl\lvert(-1)^na_n\bigr\rvert=0$.

Answer 2

If a series $\sum_nb_n$ converges, then $b_n\to 0$. Now, let $b_n = (-1)^na_n$. If $(a_n)$ doesn't tend to zero, then $(b_n)$ doesn't. So, your series cannot converge.

Answer 3

Yes it is divergent since

• for $n=2k \implies (-1)^{2k}a_{2k} \to \infty$

• for $n=2k+1 \implies (-1)^{2k+1}a_{2k+1} \to -\infty$