$$

\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.

You are currently browsing the monthly archive for **April 2020**.