I think I got it. It turns out that the series expansion of h doesn't have any constant term, so by some theorem g(h(x)) has a series expansion with strictly positive radius. To calculate the terms up to i only need to consider the first 3 terms of (where the are the coefficient of the series expansion of g) since all other terms have powers of x greater than 5 and then collect terms...

The next question is showing that the radius of convergence cannot exceed Pi. I showed that f(x) is not differentiable at x=Pi using the definition of the derivative.

Does any of this make sense to you?