Let $a = \sum_k a_k X^k \in \mathbb C [\![ X ]\!]$ with $a_0 = 1$ and convergence radius $\rho_a > 0$. I want to show that the convergence radius of the inverse $b = \sum_k b_k X^k \in \mathbb C [\![ X ]\!]$ is also greater than 0. How can I do that?
Radius of convergence of the inverse of a power series
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calculus
analysis
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2are you looking at the inverse or at $1/a$ (the reciprocal)? – 2012-06-15
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0Hi @Thomas: I know you can answer both questions! Why don't you undelete your answer? – 2012-06-15
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0It is my impression the OP does not know that holomorphic functions admit a power series representation but is looking for an answer using sequence manipulation. That I could, theoretically, also provide, but it is rather elaborate. Sure, I can undelete it. – 2012-06-15
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0I'm looking for the inverse in the ring of formal power series, i.e. a $b = \sum_k b_k X^k$ such that $a b = 1$, where $1 = c = \sum_k c_k X^k$ means, that $c_0 = 1$ and $c_k = 0$ for $k \geq 1$. It's easy to find these $b_k \in \mathbb C$. But is it possible to argue directly for example with the formula from Hadamard for the convergence radius of power series? – 2012-06-15