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?
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$\begingroup$ You mean proving the functional equation $f(x)f(y) = f(x+y)$? $\endgroup$– mvwJul 10, 2014 at 19:56
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1$\begingroup$ ... but the derivative tricks are much more fun $\endgroup$– Hagen von EitzenJul 10, 2014 at 20:33
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1$\begingroup$ ^ Yes and yes. I suppose it's not that bad, I thought it was worse. $\endgroup$– user82004Jul 10, 2014 at 21:01
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1 Answer
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$$\begin{align} \exp{(a+b)}&=\sum_{n=0}^{\infty}\frac{(a+b)^n}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{a^kb^{n-k}}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{a^kb^{n-k}}{k!(n-k)!}\\ &=\left(\sum_{k=0}^{\infty}\frac{a^k}{k!}\right)\cdot\left(\sum_{n=0}^{\infty}\frac{b^n}{n!}\right)\\ &=\exp{(a)}\cdot\exp{(b)}.~~\blacksquare \end{align}$$
The key steps of the derivation are the Binomial Theorem and the Cauchy product formula.
