$X_{n}$ is a random variable, which is discrete uniformly distributed with $\mathbb{P}(X_{n}=\frac{k}{n})=\frac{1}{n}$, for $k \in \{1,...,n\}$. There is also given a random variable $X$, with $X\sim U[0,1]$. I have to show that $X_{n}$ converges to $X$ in distribution using moment methods.
My ideas was: Trying to show straightaway, that $\lim_{n\to \infty} \mathbb{E}[X_{n}^{k}]=\mathbb{E}[X^{k}]$. I tried to rewrite it per definition \begin{align} \lim_{n\to \infty}|\mathbb{E}[X_{n}^{k}]-\mathbb{E}[X^{k}]|=\lim_{n\to \infty}\Biggl|\frac{1}{n}\sum_{k=1}^{n}\Biggl(\frac{k}{n}\Biggr)^{p}-\frac{1}{p+1}\Biggr|=\lim_{n\to \infty}\Biggl|\frac{1}{n^{p+1}}\sum_{k=1}^{n}k ^{p}-\frac{1}{p+1}\Biggr|=...=0 \end{align} Where after the first equals sign I apply definition of expected value for discrete uniform distributed random variable and calculate a moment for continuous uniform distribution. I tried to find some estimation of the summation $\sum_{k=1}^{n}k ^{p}$, but the formula I found seemed to be to complicate to be the right one. If someone could tell me, if I am on the right way or give me some tips how to solve this problem, I would be grateful!