Given $\sum^\infty_{k=1}\frac{(-1)^{k+1}}{\sqrt{k}}$ converges:

Question

Assume $\sum^\infty_{k=1}a_k$ and $\sum^\infty_{k=1}b_k$ are both non-absolutely convergent. Find specific examples of $\sum^\infty_{k=1}a_k$ and $\sum^\infty_{k=1}b_k$ such that $\sum^\infty_{k=1}(a_kb_k)$ is not convergent.

Answer 1

As noticed by weee in the comments, we can just use the series in the title both for $a_k$ and $b_k$.

More in general we can use any

$$\sum^\infty_{k=1}a_k=\sum^\infty_{k=1}\frac{(-1)^{k+1}}{k^a} \quad \quad \sum^\infty_{k=1}b_k=\sum^\infty_{k=1}\frac{(-1)^{k+1}}{k^b}$$

with

• $0<a,b<1$
• $0<a+b<1$