Let $(X,\|\cdot\|)$ be a normed linear space. Recall from prior results that $(X,\|\cdot\|)$ is Banach $\iff$ any absolutely convergent series in $(X,\|\cdot\|)$ converges.
(a) Give an example of a Banach space $(X,\|\cdot\|)$ and a convergent series which is not absolutely convergent.
(b) Give an example of a normed linear space $(X,\|\cdot\|)$ and an absolutely convergent series which is not convergent.
I haven't tried much, I really don't know where to begin.