= Solution
Write $q(x)=\sup_k|\sum_{n=1}^kx_n|$. This is indeed a <norm>: the <triangle inequality> and homogeneity follow from the scalar absolute value, and vanishing of every <partial sum> forces every coordinate to vanish. It is finite on $\ell^1$ because $q(x)\leq\|x\|_1$.
Let $a_n=(-1)^{n+1}/n$ and let $a^{(N)}$ be its first $N$ entries followed by zeros. The alternating-<series> bound gives, for $M>N$,
$$
q(a^{(M)}-a^{(N)})\leq\frac1{N+1}.
$$
Thus these truncations are Cauchy for $q$. But $|x_n|\leq2q(x)$ for every coordinate, since $x_n$ is the difference of two consecutive <partial sums>. Any $q$-limit would consequently have coordinates $a_n$. This sequence is not in $\ell^1$. Hence
$$
\boxed{(\ell^1,q)\text{ is not a Banach space}.}
$$
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