Generating function for $1,\frac{1}{2!},\frac{1}{4!},\dotsc$

Question

I'm trying to find a generating function for $1,\frac{1}{2!},\frac{1}{4!},\dotsc,\frac{1}{(2k)!},\dotsc$. This means I want to get a function represented by $1+\frac{x}{2!}+\frac{x^2}{4!}+\dotsb$. This looks close to $\frac{e^x+e^{-x}}{2}$, but it has the wrong powers. Can I please get a hint? I know it must have something to do with $e^x$ but none of the things I try seem to work out.

Answer 1

Just change x to x^0.5.. So it will give your req series..