Exponential series/Real/Functional equation/Fact/Proof

Proof

The Cauchy product of the two exponential series is

$\sum _{n=0}^{\infty }c_{n},$

where

${}c_{n}=\sum _{i=0}^{n}{\frac {x^{i}}{i!}}\cdot {\frac {y^{n-i}}{(n-i)!}}\,.$

This series is due to fact absolutely convergent and the limit is the product of the two limits. Furthermore, the ${}n$-th summand of the exponential series of ${}x+y$ equals

${}{\frac {(x+y)^{n}}{n!}}={\frac {1}{n!}}\sum _{i=0}^{n}{\binom {n}{i}}x^{i}y^{n-i}=c_{n}\,,$

so that both sides coincide.

To fact