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.