April 2020

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Cute math, $\exp(x)\cdot\exp(-x)=1$

$$\exp(x)\cdot\exp(-x)$$
$$=\sum_{n=0}^\infty \frac{x^n}{n!}\cdot\sum_{n=0}^\infty \frac{(-x)^n}{n!}$$
$$=\sum_{n=0}^\infty \frac{x^n}{n!}\cdot\sum_{n=0}^\infty(-1)^n \frac{x^n}{n!}$$
$$=\sum_{n=0}^\infty\sum_{i=0}^n (-1)^i \frac1{i!}\frac1{(n-i)!}x^n \quad\mathrm{collecting\ coef\ of\ } x^n$$
$$= 1+ \sum_{n=1}^\infty \frac{ (1-1)^n}{ n!} x^n = 1$$
using the binomial theorem on the second to last equality.