1. The problem statement, all variables and given/known data Prove that 1 + 1 + 1/2! + 1/3! +1/4! + 1/5!... + 1/n! Limit of this as n goes to infinity is equal to e. 2. Relevant equations I already showed (1 + 1/n)^n = e 3. The attempt at a solution My proof is as follows: e = limit (n to infinity) Sum (k=1 to n) of: (n!/(n-k)!) * 1/(k!*n^n) = limit (n to infinity) of: 1 + n/n + n(n-1)/n^2 * 1/2! + n(n-1)(n-2)/n^3 * 1/3! + ......... Note: each term has the same leading degree and coefficient, meaning each terms tends to 1 * 1/k! as n approaches infinity thus this entire series tends to: 1 + 1 + 1/2! + 1/3! + 1/4!.... My main question/concern is the fact that I passed the limit through the summation. It this ok? I know that it works for finite sums: ie: Limit (X_n + Y_n) = Limit(X_n) + Limit(Y_n) but I'm unsure if this is true for infinite sums. Any help would be appreciated.