# Specify whether the series $\sum_{n=n_0}^{+ \infty } a_n \cdot b_n$ must be convergent in cases a) or b)

Specify whether the series $$\sum_{n=n_0}^{+ \infty } a_n \cdot b_n$$ must be convergent when: a) $$\sum a_n$$ convergent, $$\sum b_n$$ convergent b)$$\sum a_n$$ convergent absolutely, $$\sum b_n$$ convergent. I need to check my reasoning and I need help to guide them further: If $$\sum b_n$$ convergent, then in particular a sequence of partial sum $$\sum b_n$$ is convergent, so the second condition of Dirichlet's test is met. That is why we should take care of $$a_n$$. The first condition of Dirichlet's test is $$a_n$$ monotonic and $$a_n$$ convergent to $$0$$. So I think in the case of a) the series $$\sum_{n=n_0}^{+ \infty } a_n \cdot b_n$$ not always is convergent and b) must be convergement, but there are only my thoughts and I do not know how to prove it.

$$a_n=b_n=\frac{(-1)^n}{\sqrt n}$$
For the second and since $$\sum b_n$$ is convergent so for sufficiently large $$n$$ we get $$\vert b_n\vert \le 1$$ hence $$\vert a_n b_n\vert\le \vert a_n\vert$$ hence the series $$\sum a_n b_n$$ is absolutely convergent.