I am reading the following lemma from Harary and Palmer, and I have trouble fully understanding it.

In this context, $T(x)=x\exp\left(\sum_{i\ge 1}\frac{1}{i}T(x^i)\right)$, and its radius of convergence is denoted as $\eta$.

I can follow the proof up to $b_0=\lim_{x\rightarrow \eta^-}T(x)$. However, I don't understand why this would guarantee the convergence of T(x) at $\eta$, since a power series is not necessarily continuous at its radius of convergence. Am I missing something simple? Any explanation is appreciated. Thank you!

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  • $\begingroup$ This follows from Abel's theorem on the convergence of power series at boundary points of their interval of convergence. Keith Conrad has some notes on the subject which Google should find for your easily. $\endgroup$ – Mariano Suárez-Álvarez Jan 16 '17 at 3:27
  • $\begingroup$ @MarianoSuárez-Álvarez Thank you! I am still a little confused because I'm not sure if Abel's theorem applies here. Abel's theorem assumes convergence at the radius, and shows the series is continuous. Here I know the series's left limit exists, and I want to show the series converges at the radius. So is the converse of Abel's theorem still true? I read Keith Conrad's notes and it seems his Corollary 2 applies to my case, but the proof still requires assuming the series converge at the radius. $\endgroup$ – Xiaojing Jan 16 '17 at 3:48

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