Rewriting an Analytic function based on its Power Series
So a question states: given a function f(z) that's analytic at the origin with f(0)=f '(0)=0, prove that f(z) can be written f(z)= z^2 g(z), where g(z) is analytic at z=0.
I thought this would come right out after I established that since f is analytic at the origin, and thus analytic on some open domain about the origin, it can be represented by its Maclaurin series on that domain, which has two leading zero terms. The Maclaurin representation can then be reworked to f(z)=z^2 Sum(k:[0,Inf) (f^(k)(0)*z^(k-2))/k!)
If all is right so far: how do I know that this new summation uniformly converges to anything? (uniform convergence to ensure that it is analytic) I kind of intuit that it's really the coefficients that make it converge and a finite offset of the exponents doesn't effect much of anything, but I'm not sure, and I wouldn't know how to prove it.
Any hints would be appreciated.