Hi,

Here's the question: Consider the functionfdefined uniquely by the relation

$\displaystyle (f(x))^3=x^2sin(x)$

Show thatfhas a series expansion in powers ofxwith strictly positive radius. (Hint: consider $\displaystyle g(h(x))$ where $\displaystyle g(y)=(1+y)^\frac{1}{3}$ and $\displaystyle h(x)=\frac{sin(x)-x}{x}$)

So far I figured out that $\displaystyle f(x)=x g(h(x))$, 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 forfas far as the term in $\displaystyle x^5$, 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...