How to show by using Power series :
Proof that sum ((a+b)^n)/n!=(sumsubk ((a^k)/k!))(sum subm((b^m)/m!)
For this, you need to know the Cauchy product theorem, which says that if and are absolutely convergent series then their product is given by .
If you apply that to the power series for exp(a) and exp(b) then you get . We want to show that this is equal to . Comparing the two series, you see that this means showing that . But if you multiply both sides by n!, that is just the binomial expansion of .