Algebraic Proof of Sum of Exponential Powers is Product of Exponentials

Question

Can somebody provide a proof of the summation of powers law for the product of two exponentials, using only algebra and the Taylor series, no derivatives or calculus tricks?

Answer 1

$$\begin{align} \exp{(a+b)}&=\sum_{n=0}^{\infty}\frac{(a+b)^n}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{a^kb^{n-k}}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{a^kb^{n-k}}{k!(n-k)!}\\ &=\left(\sum_{k=0}^{\infty}\frac{a^k}{k!}\right)\cdot\left(\sum_{n=0}^{\infty}\frac{b^n}{n!}\right)\\ &=\exp{(a)}\cdot\exp{(b)}.~~\blacksquare \end{align}$$

The key steps of the derivation are the Binomial Theorem and the Cauchy product formula.