Absolutely convergent but not convergent

Here, Lemma $2.1$ states that

A normed space $X$ is complete if and only if every absolutely convergent series is convergent.

I would like to know a series which is absolutely convergent but not convergent. Can someone give such example? I always thought that absolutely convergent series implies the series converges.

If the space is not complete, absolute convergence actually doesn't imply convergence. Take for example $X = \mathbb{Q}$ (with the norm inherited from $\mathbb{R}$) and choose a sequence of rational numbers $a_n$ such that $$ \sum_{n=1}^\infty a_n $$ is irrational, but $$ \sum_{n=1}^\infty |a_n| = 1. $$

See this question for details. Thus, in $X$, the series is absolutely convergent, but not convergent.