How to bound this series

Question

I am currently reading through Tsybakov's Introduction to Nonparametric Statistics, and in it, he uses the estimate $$\left(\sum_{m=n}^\infty (m-1)^{-2\beta}\right)^{1/2} = O\left(n^{-\beta + 1/2}\right)$$ (here $\beta > 0$ if that is important) or in other words, $$\sum_{m=n}^\infty (m-1)^{-2\beta} = O\left(n^{-2\beta + 1}\right)$$ o more abstractly, one might even prove that (with proper assumptions on $k$ if necessary) $$\sum_{m=n}^\infty (m-1)^{-k} = O\left(n^{-k + 1}\right)$$ This is a bit different than what I usually encounter, the trickiness arising because the summation index is in the base, not the exponent. How does one go about proving such an estimate?