Personal reference on power series for some commonly used functions.
What counts as “common” is somewhat arbitrary.

\[
e^x = \sum_{n=0}^\infty \frac{x^n}{n!}
= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
\]

\[
\sin x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} x^{2n+1}
= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots
\]

\[
\cos x = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^{2n}
= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots
\]

## See also