Prove if series converges or diverges

**Pinkk** · March 3rd 2010, 05:19 PM

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

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

**TheEmptySet** · March 3rd 2010, 05:24 PM

Originally Posted by Pinkk

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

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

$\sum_{n=1}^{\infty}\frac{n-1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n^2}$

The first sum is the harmonic series and diverges to infintiy and the 2nd sums to $\frac{\pi^2}{6}$ is I remember correctly so the sum diverges to infinity.

**skeeter** · March 3rd 2010, 05:28 PM

Originally Posted by Pinkk

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

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

limit comparison with the known divergent series $\sum \frac{1}{n}$ ...

$\lim_{n \to \infty} \frac{\frac{n-1}{n^2}}{\frac{1}{n}}$

$\lim_{n \to \infty} \frac{n-1}{n^2} \cdot \frac{n}{1} = 1$

series diverges

**Pinkk** · March 3rd 2010, 05:29 PM

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.

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