Expectation of the min(X1...Xn) in the discrete case

Question

Given $n$ i.i.d. uniformly random variables $X_{1},..X_{n}$, each drawn from $\{1,\dots,m\}$, I am trying to compute the expectancy of the minimum. This seems to be rather easy to solve in the continuous case. Nevertheless, in the discrete case I define $Y$ to be $\min(X_{1},..X_{n})$ and then I observe that $\Pr(Y>y)=(\frac{m-y}{m})^{n}$. The expectancy is then $\sum_{y=0}^{m-1}P(Y>y)=\sum_{y=0}^{m-1}(1-\frac{y}{m})^n$, but it is not clear to me how to work on this in order to obtain something in terms of $n$ and $m$. Any ideas on the best approach to this?