# Thread: comparison test sum exists

1. ## comparison test sum exists

I know that:

this sequnce $\sum _{k=1}^{\infty }{k}^{-1}$ doesnot exist

but this sequence $\sum _{k=1}^{\infty }-{k}^{-2}$does exist

How can i prove this sequence $\sum _{k=1}^{\infty }{{k}^{-1}-{k}^{-2}}$ does not exists

2. $

\sum _{k=1}^{\infty }\left( {{k}^{-1}-{k}^{-2}} \right)=\sum_{k=1}^{\infty} \frac{1}{k}-\sum_{k=1}^{\infty} \frac{1}{k^2}
$

We know that $\sum_{k=1}^{\infty} \frac{1}{k} \rightarrow \infty$ and $\sum_{k=1}^{\infty} \frac{1}{k^2}=\frac{\pi^2}{6}$

so the RHS approaches $\infty$ meaning that the series does not converge.