I am working through some textbook problems and came across a problem I am having difficulty with. The problem asks to give the bias of a point estimate, namely for a given set of data $X = \{x_1, x_2,\ldots,x_n\}$ drawn from a discrete $\text{Uniform}\{0,1,\ldots, k\}$, what is the bias of the estimator $\hat\theta = \max(X)$?
I'm pretty sure I have a representation for $E[\hat\theta]$, namely:
$$E[\hat\theta] = \sum_{i=0}^{n-1} \left(\frac{k-1}{k}\right)^{ni} \left(1-\left(\frac{k-1}{k}\right)^n\right)(k-i)\\ \hspace{.91cm} = \left(1-\left(\frac{k-1}{k}\right)^n\right)\sum_{i=0}^{n-1} \left(\frac{k-1}{k}\right)^{ni} (k-i)$$
The first term represents the probability that the numbers larger than $k-i$ have not been drawn, while $1 - (\frac{k-1}{k})^n$ represents the probability of any number in the range being drawn. Thus, those terms together represent the probability that $k-i$ is the largest number in the sample. However, to measure the bias, I need to simplify the summation as bias is $E[\hat\theta] - k$, but I am a bit unsure how to simplify the summation.