upper bound of p-series

Question

I saw the following relationship used in a couple of papers: $\sum_{t=2}^\infty \frac{1}{t^\alpha}\le \frac{1}{\alpha-1}$ for $\alpha>1$. Can anyone explain how this relationship is obtained? Thanks.

Answer 1

If you're familiar with the integral test, you have probably seen inequalities like this: $$ \sum_{t=2}^\infty t^{-\alpha}\le\int_1^\infty t^{-\alpha},\ \ \ \alpha>1 $$ Does the argument make sense from here?