Series in a space which is not complete

Question

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with

$$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$

which means there exists no $x \in X$ with

$$\lim_{n \to \infty} \left\|x-\sum_{k=1}^{n} x_k \right\|=0. $$

Can anybody give me an example of such a sequence $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}}$?

I know that this situation is only possible if $X$ is not complete.

Answer 1

Take $X=c_{00}$---the space of all sequences which are almost everywhere $0$ and as $x_n$---the sequence having $\frac{1}{2^n}$ on $n$-th place and $0$ elsewhere.