f(x)=cos2xとする。
f'(x)=-2sin2x,f''(x)=-\(2^{2}\)cos2x,\(f^{3}\)(x)=\(2^{3}\)sin2x,\(f^{4}\)=\(2^{4}\)cos2x,...
f'(0)=0,f''(0)=-\(2^{2}\),\(f^{3}\)(0)=0,\(f^{4}\)(0)=\(2^{4}\)
cos2x=1-\(2^{2}\)*\(x^{2}\)/2!+\(2^{4}\)*\(x^{4}\)/4!-\(2^{6}\)*\(x^{6}\)/6!+\(2^{8}\)*\(x^{8}\)/8!-...
=Σ(-1\()^{n}\)*2^(2n)*x^(2n)/(2n)! [n=0..∞]

g(x)=log{(1+x)/(1-x)}と解釈すると。
g(x)=log(1+x)-log(1-x)
g'(x)=1/(1+x)+1/(1-x)={1-x+\(x^{2}\)-\(x^{3}\)+...}+{1+x+\(x^{2}\)+\(x^{3}\)+...}
=2{1+\(x^{2}\)+\(x^{4}\)+...}
=2Σx^(2n) [x=0..∞]