1. Variance problem

I am working my way through Feller (3rd ed, vol. 1), and I got stuck on the following problem (IX.9 problem 19a):
A man with n keys wants to open his door and tries the keys independently and at random. Find the mean and variance of the number of trials if unsuccessful keys are not eliminated from further selection. (Assume that only one key fits the door.)
Computing the mean presented no difficultly. I've worked out the variance to be $\displaystyle \sum_{k=1}^{\infty}{k^2 {{(n-1)^{k-1}}\over{n^k}}}-n^2$. However, I am not sure how to compute the sum, and would appreciate any hints.

2. Originally Posted by aix
I am working my way through Feller (3rd ed, vol. 1), and I got stuck on the following problem (IX.9 problem 19a):
A man with n keys wants to open his door and tries the keys independently and at random. Find the mean and variance of the number of trials if unsuccessful keys are not eliminated from further selection. (Assume that only one key fits the door.)
Computing the mean presented no difficultly. I've worked out the variance to be $\displaystyle \sum_{k=1}^{\infty}{k^2 {{(n-1)^{k-1}}\over{n^k}}}-n^2$. However, I am not sure how to compute the sum, and would appreciate any hints.
Let X be the random variable number of trials before getting the correct key.

X follows a geometric distribution with p = 1/n.

You'll find the derivations of mean and variance here: http://www.win.tue.nl/~rnunez/2DI30/...tributions.pdf

But personally I think it's easier to calculate and then use the moment generating function.