$\displaystyle \sum\,\frac{n-1}{n^{2}}$
I'm having a brain fart, I don't know if this converges or not.
$\displaystyle \sum_{n=1}^{\infty}\frac{n-1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n^2}$
The first sum is the harmonic series and diverges to infintiy and the 2nd sums to $\displaystyle \frac{\pi^2}{6}$ is I remember correctly so the sum diverges to infinity.