The Taylor series of #f(x)=cosx# at #x=0# is

#f(x)=sum_{n=0}^infty (-1)^nx^{2n}/{(2n)!}#.

Let us look at some details.

The Taylor series for #f(x)# at #x=a# in general can be found by

#f(x)=sum_{n=0}^infty {f^{(n)}(a)}/{n!}(x-a)^n#

Let us find the Taylor series for #f(x)=cosx# at #x=0#.

By taking the derivatives,

#f(x)=cosx Rightarrow f(0)=cos(0)=1#

#f'(x)=-sinx Rightarrow f'(0)=-sin(0)=0#

#f''(x)=-cosx Rightarrow f''(0)=-cos(0)=-1#

#f'''(x)=sinx Rightarrow f'''(0)=sin(0)=0#

#f^{(4)}(x)=cosx Rightarrow f^{(4)}(0)=cos(0)=1#

Since #f(x)=f^{(4)}(x)#, the cycle of #{1,0,-1,0}# repeats itself.

So, we have the series

#f(x)=1-{x^2}/{2!}+x^4/{4!}-cdots=sum_{n=0}^infty(-1)^n x^{2n}/{(2n)!}#