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.

Try $e^{\sqrt{x}}$ rather than $e^x$. – rogerl Feb 20 '15 at 01:30 — rogerl, Feb 20 '15 at 01:30
Thanks! That was exactly what I needed. – bret Feb 20 '15 at 01:36 — bret, Feb 20 '15 at 01:36

score 1 · Accepted Answer · answered Feb 20 '15 at 08:13

1

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

answered Feb 20 '15 at 08:13

Shakul Pathak

463

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

1 Answers1