Let $n$ be a positive integer, $\beta > 1$, and let $X$ be a random variable uniformly distributed over $\{0, \ldots , n -1\}$. Show that $\mathbb{E}[X^\beta] \leq n^\beta / (\beta + 1)$.
I don't know how to get started with this. Can someone help me out?
If $X$ is uniformly distributed over $\{0,\ldots,n-1\}$ then you can calculate the expectation: $$\mathbb{E}[X^\beta] = \frac{1}{n} \sum_{i=0}^{n-1} i^\beta. $$ Now there are any number of ways to proceed. For example, you could estimate the sum with an integral.