## 8.4. Taylor Series

### Proposition 8.4.12: Taylor Series for the Cosine Function

Taking derivatives at *c=0* we quickly see that all odd
derivatives at *c=0* are zero while the even ones are plus or
minus one. Thus:

cos(x) = 1 - 1/2! x^{2}+ 1/4! x^{4}- ... + (-1)^{n}/(2n)! x^{2n}+ R_{2n+2}(x)

We need to show that the remainder *R _{2n+2}* goes to zero
as

*n*goes to infinity. Let's use the Lagrange remainder with

*c=0*:

But for a fixed *x* we have
already shown before that for
any fixed number *x*

^{|x|n}/_{n!}= 0

which finishes the proof.

### Fun Facts:

*cos(0) = 1**cos*is an even function, i.e.*cos(-x) = cos(x)**cos(x - /2) = -sin(x)**cos(x) = - sin(x)*

The fun facts are all easy and are left as exercises. For the third fact you
might start with *f(x)=sin(x)* and develop that in a series around
*c = /2*.