 Solution

Let $e_j$ be the coordinate vectors and define finitely supported vectorsEach $v_k$ belongs to ${\ell }^1$, and $\Vert v_k{\Vert }_{\infty }=2^{-k}$. Hence ${\sum }_k{v}_k$ is absolutely convergent for the supremum norm. Its partial sums converge uniformly in ${\ell }^{\infty }$ to the sequence $v$ whose entries on the $k$th block all equal $2^{-k}$. But every block contributes one to its ${\ell }^1$ sum:Thus $v\notin {\ell }^1$. Any supremum-norm limit in ${\ell }^1$ would have the same coordinates as $v$, an impossibility. The partial sums are Cauchy and have no limit in the space. Therefore $({\ell }^1,\Vert \cdot {\Vert }_{\infty })$ is not Banach, and the displayed series is the required absolutely convergent but nonconvergent series in that space.