Solution
= Solution
Fix $a$ and define $g(b)=e^{a+b}e^{-b}$ using only the power <series>. Termwise <differentiation> gives $(e^z)\prime=e^z$, and the product rule gives $g\prime(b)=0$. Hence $g$ is constant, so $g(b)=g(0)=e^a$. Multiplication by $e^b$ yields $e^{a+b}=e^ae^b$.