How do you prove ex * ey = ex + y using series?

[lalalalizzay]

How do you prove e^x * e^y = e^{x + y} using series?



Prove:

suppose sum a_n and sum b_n converge absolutely then the series c_n with c_n = sum (from k=0 to n) a_k*b_n-k converges absolutely to the limit sum a_n * sum b_n

Thank you!

Or if you have seen either of these on here could you redirect me to that page

 

[Tac-Tics]

For the first one, what is the formula for e^x? That should make the proof obvious.

For the second one, I don't know, and dinner's ready, but it probably follows from using the epsilon-delta definition of a limit.

 

[Dick]

Hint: (a0+a1)*(b0+b1)=a0*b0 + (a0*b1+a1*b0) + a1*b1. In the first term the indices sum to 0, in the second to 1 and in the last to 2. Do you see how that is related to what you are asking?