$\displaystyle \sum\,\frac{n-1}{n^{2}}$

I'm having a brain fart, I don't know if this converges or not.

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- Mar 3rd 2010, 05:19 PMPinkkProve if series converges or diverges
$\displaystyle \sum\,\frac{n-1}{n^{2}}$

I'm having a brain fart, I don't know if this converges or not. - Mar 3rd 2010, 05:24 PMTheEmptySet
$\displaystyle \sum_{n=1}^{\infty}\frac{n-1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n^2}$

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. - Mar 3rd 2010, 05:28 PMskeeter
- Mar 3rd 2010, 05:29 PMPinkk
Hmm, that seems obvious enough, but since I'm in a real analysis course, I have to show this a bit more formally; it was never proven in my class that the sum/difference of a divergent and a convergent series is divergent.