Exponential series/Real/Functional equation/Fact/Proof

The Cauchy product of the two exponential series is

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

where

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

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

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

so that both sides coincide.