Composition of two power series

I thought of this question:

$f, g$ are functions that could be written as a power series in a disc with radius $R>0$, i.e. $$\forall |x|<r, f(x)=\sum a_n x^n, g(x)=\sum b_n x^n$$. If $b_0 = 0$, do we always have that $f(g(x))$ could be written as a power series in a disc containing 0?

I can't prove or find a counterexample, could anyone give me a hand?

Thanks a lot~

• Is some complex analysis acceptable, or do you want a real-methods-only proof? – Daniel Fischer May 27 '17 at 13:52
• @DanielFischer I'd like to be in a complex plane :) – pqros May 27 '17 at 14:00
• Then: the composition of holomorphic functions is holomorphic; and a function is holomorphic if and only if it is (complex) analytic. – Daniel Fischer May 27 '17 at 14:02
• We can express the composition of $f (g (x))$ as a *formal power series* when $b_0=0$, but this does not tell us the radius of convergence for the result. – hardmath May 27 '17 at 14:06
• @DanielFischer Oh I think I should reread my complex analysis, thanks~ – pqros May 27 '17 at 14:09

$f (x)$ exists if $|x|<R$
$f (g (x))$ exists if $|g (x)|<R$.
Since $g (0)=0$, there exists $\eta>0$ such that
$g ((-\eta,\eta))\subset (-R,R)$.
thus $f (g (x))$ exists for $|x|<\eta.$