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Let $ f(z) $ and $ g(z) $ be power series, and assume that $$ f(z) = \frac{1+z g(z)}{1-z} $$ Can anyone show that $ f $ and $ g $ have the same radius of convergence? I cannot figure this out. This came up in a paper I was reading but the author just stated it.

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  • $\begingroup$ That surely can't be true if the RoC of $g$ is larger than $1$ and $g(1)\neq -1$, since otherwise $f$ has a singularity at $z=1$, and hence the RoC of $f$ is at most $1$? $\endgroup$ – Chappers Sep 18 '15 at 0:00
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This is not true. Counterexample is $g(z)=0$ (constant) which has radius of convergence $\infty$, in which case $f(z) = 1/(1-z)$ has radius of convergence $1$. (This is assuming that your center is $a=0$, but no matter what the center is, this would be a counterexample.)

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