Is $\ell^1$ complete with this norm?

For $ x \in \ell^1$ we set $ ||x|| = \sup\limits_{N \in \mathbb{N}}|\sum\limits_{n=1}^{N}x_n|$ .
One can easily see that this is a norm on $ \ell^1$ . I was wondering if this space is now complete, so I tried finding an absolute convergent series that does not converge, but did not find anything.
Thx in advance!