I'm trying to add two power series which are given as two functions $$f(x)=\sum_{n=0}^\infty \frac{n!}{n^n}x^n $$ $$g(x)=\sum_{n=0}^\infty (-1)^n\frac{n!}{n^n}x^n $$
I can't seem to find the power series of: $$\frac{1}{2}(f(x)+g(x))$$
I have no idea what I'm doing and attempted: $$\frac{(\frac{n!}{n^n}x^n+(-1)^n\frac{n!}{n^n}x^n)}{2}$$
Which simplifies to but isn't quite right: $$\frac{n!((-1)^n+1)x^n}{2n^n}$$
Any help and guidance on the right path would be so very appreciated.