An upper bound of the sum $\sum_{n\le X}n^{k-2}$

Question

Let $k$ be an integer $\ge 5,$ I'm concerned with finding an upper bound of the following sum: $$\sum_{n\le X}n^{k-2}$$ Any help would be appreciated.

Answer 1

Here is one simple way to bound that summation. $$\displaystyle\sum_{n = 0}^{X}n^{k-2} \le \sum_{n = 0}^{X}\int_{n}^{n+1}x^{k-2}\,dx = \int_{0}^{X+1}x^{k-2}\,dx = \dfrac{1}{k-1}(X+1)^{k-1}$$