Taylor Series

The Taylor series of a function $f$ about a point $a$ is

$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x - a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$

When $a = 0$ the series is called a Maclaurin series.

Famous expansions:

• $e^x = \sum \frac{x^n}{n!}$
• $\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$
• $\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}$
• $\frac{1}{1-x} = \sum x^n$ (for $|x| < 1$).

Truncating the series at degree $n$ yields a polynomial approximation. This is how calculators compute trig and exp internally and how physics approximates "small angle" or "low velocity" behaviour. The Taylor series exists wherever the function is infinitely differentiable and the remainder term tends to zero.