Thread: Prove if series converges or diverges

1. Prove 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.

2. Originally Posted by Pinkk
$\displaystyle \sum\,\frac{n-1}{n^{2}}$

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.

3. Originally Posted by Pinkk
$\displaystyle \sum\,\frac{n-1}{n^{2}}$

I'm having a brain fart, I don't know if this converges or not.
limit comparison with the known divergent series $\displaystyle \sum \frac{1}{n}$ ...

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

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

series diverges

4. 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.