While doing my statistical mechanics homework today I arrived at a sum that looked like Sums like this come up frequently when you're working with random walks, or flipping coins, or counting states of quantum mechanical systems where only two energies are allowed, or in any other number of contrived situations. It's almost a good looking sum because everything to the right of the factor of m looks like the sum for a binomial expansion : which simply evaluates to: It turns out that there's an easy way to get the factor of m out of the sum and get on with your life! First notice that: so that the sum can be re-written as: So, we rather handily got rid of the factor of m. The extra factor of p can be taken outside of the sum since it has nothing to do with the summation index m. Furthermore, the order of summation and differentiation can be interchanged to arrive at: Now the sum actually is a binomial expansion and after simplifying and performing the derivative that was i...