1. The problem statement, all variables and given/known data Self study, Bransden and Joachain, Quantum Mechanics, problem 5.8, as written above in title, c a complex number, A and B matrices. I found the statement itself on Wikipedia but no proof. 2. Relevant equations I've used power series to prove e^(A+c)=e^A*e^c, and I checked [A-a,B-b]=[A,B] I've written a lot of products of power series. 3. The attempt at a solution Notation I use: C^n = (A+B)^n respecting order, c^n = (a+b)^n with all A's before B's. e.g.: (A+B)^2 = AA + AB +BA + BB, and (a+b)^2 = AA + 2AB + BB (A+B)^0 = I (A+B)^1 = A+B (A+B)^2 = (a+b)^2 - k (A+B)^3 = (a+b)^3 - 3k(A+B) (A+B)^4 =(a+b)^4 - 6k(a+b)^2 + 3k^2 (A+B)^5 =(a+b)^5 - 10k(a+b)^3 + 15k^2(a+b) (A+B)^6=(a+b)^6 - 15k(a+b)^4 + 45k^2(a+b)^2 - 15k^3 I have more but error checking takes forever. I can tell the second coefficient is (n choose 2) and I have more partial patterns, but not the entire pattern figured out. I've tried power series written in lots of ways and made no progress. Since this was just one problem at the end of a chapter I expect there is a simple solution I am missing, but I have really hunted for a very long time, and I don't see how to make this happen, although the terms I have worked out make the theorem look fairly plausible. Any tips or pointers welcome.