How Do You Prove f^(2n)(0)=(2n)!/(n!) for f(x)=e^(x^2)?

[leftwing1018]

Here's the problem:

If f(x)=e^(x^2), show that f^(2n)(0)=(2n)!/(n!).

Really I don't even know where to begin. Any help on where to start would be great.

 

[dx]

Hi leftwing1018,

Welcome to PF. Here's a hint: write f(x) as an infinite series using the well-known series expansion of e^x, and then compare that with the Taylor expansion of a general function near x = 0.