To show $\zeta(s)$ is convergent (click here and go to page 1), it is written in Definition 16.1. that-
use the integral test on an open ball strictly to the right of the line $Re(s) = 1$
Now, I understand that $\zeta(s)=\sum_{n \geq 1}n^{-s}$ is convergent when $s=2$ from this online tutorial of integral test using graph, I assume similary we can prove for $s>2$, $\zeta(s)=\sum_{n \geq 1}n^{-s}$ is convergent. But I cannot relate this with integral test on an open ball strictly to the right of the line $Re(s) = 1$.
Can any one plese elaborately explain how can we use integral test on an open ball strictly to the right of the line $Re(s) = 1$?
Thanks.