
# 4.1: Taylor Series


One series you have encountered before is Taylor’s series,

$f(x) = \sum_{n=0}^{\infty} f^{(n)}(a)\frac{(x-a)^n}{n!}, \label{eq:IV:taylor}$

where $$f^{(n)}(x)$$ is the $$n$$th derivative of $$f$$. An example is the Taylor series of the cosine around $$x=0$$ (i.e., $$a=0$$),

\begin{aligned} &&\qquad&\cos(0) &= 1,\nonumber\\ \cos'(x) &= -\sin(x),&&\cos'(0)&=0,\nonumber\\ \cos^{(2)}(x) &= -\cos(x),&&\cos^{(2)}(0)&=-1,\\ \cos^{(3)}(x) &= \sin(x),&&\cos^{(3)}(0)&=0,\nonumber\\ \cos^{(4)}(x) &= \cos(x),&&\cos^{(4)}(0)&=1.\end{aligned}

Notice that after four steps we are back where we started. We have thus found (using $$m=2n$$ in ($$\PageIndex{1}$$)) )

$\cos x = \sum_{m=0}^\infty \frac{(-1)^m}{(2m)!} x^{2m},$

Exercise $$\PageIndex{1}$$

Show that $\sin x = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)!} x^{2m+1}. \nonumber$