1. The problem statement, all variables and given/known data Prove that all these equations are equal to calculate e: e1= lim (1+1/n)^n - for n to infinity e2= E 1/k! - for k from 0 to m e3= 1 / E[(-1)^k/k!] - for k from 0 to m e4= E(k^2)/2k! - for k from 1 to m e5= E(k^3)/5k! - for k from 1 to m e6= E(k^4)15k! - for k from 1 to m I used E as the symbol for summing a series of numbers. Guess it´s the most similar letter to that symbol. Hope you understand the equations...if not I have them in a word document. 2. Relevant equations MacLaurin series, binomial expansion...etc 3. The attempt at a solution e1 is done since it´s the most common definition for e. e2 is done thanks to the binomial expansion e3 have no idea e4 to e6 I think it has something to do with Bell´s number, but don´t know how to prove these are equal. I tried expanding them over several pages and finding some common factors but to no use. Can someone help me? I´ve browsed the web for days but I can only find people saying that they are ways of calculating e and not the reason why they are equal. Many thanks!