Expected number of rolls A fair m-sided dice is rolled and summed until the sum is at least N. What is the expected number of rolls? In other words what is the number of rolls if we roll a m-sided dice and the sum of rolls become at least N.

If $f(N)$ is the expected number of rolls, by conditioning on the first roll we have
$f(N) = 1 + m^{-1} \sum_{j=1}^m f(N-j)$ for $N > 0$, with $f(N) = 0$ for $N \le 0$.  The generating function is $$g(z) = \sum_n f(n) z^n = \dfrac{mz}{m - (m+1)z + z^{m+1}}$$ 
EDIT: If you're interested in the asymptotic behaviour of $f(N)$ as $N \to \infty$ for fixed $N$, you want to look at the smallest root of the denominator, which is $z=1$.  We have $$g(z) = \dfrac{2}{(m+1)(z-1)^2} - \dfrac{2(m-4)}{3(m+1)(z-1)} + h(z)$$
where $h(z)$ is analytic in a neighbourhood of the unit disk.  Corresponding to this we get
$$ f(N) = \dfrac{2 N}{m+1} + \dfrac{2(m-1)}{3(m+1)} + O(c^{-N}) \ \text{as}\ N \to \infty $$
for some $c > 1$ (depending on $m$).
