Thread: Analysis, Power Series

Hi,

Here's the question: Consider the function f defined uniquely by the relation



Show that f has a series expansion in powers of x with strictly positive radius. (Hint: consider where and )

So far I figured out that , but that's about it. I tried "Taylor-expanding" these different functions, but can't do it about 0 and so it gets complicated. Also i am not too sure about composing the functions. It seems you can do it with formal power series, but i don't know for this particular problem.

I could possibly argue the existence of the series using some theorems, but the next question ask to compute the series for f as far as the term in , so I guess I really have to find a series expansion explicitely.

I would be thankful if anyone could tell me how I should start working on this...

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?