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GeezerVison is showing mathworld.wolfram.com — Abel's Convergence Theorem -- from Wolfram MathWorld 
Abel's Convergence TheoremAbel's convergence theorem extends the uniform convergence of a power series and the continuity of its sum to a convergent boundary point. Given a Taylor series
where the complex number
Abel's theorem states that, if
Stated in words, Abel's theorem guarantees that, if a real power series converges for some positive value of the argument, the domain of uniform convergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point. See alsoAbel-Regularized Sum, Power Series, Uniform ConvergenceExplore with Wolfram|Alpha
More things to try: ReferencesArfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 773, 1985.Referenced on Wolfram|AlphaAbel's Convergence TheoremCite this as:Weisstein, Eric W. "Abel's Convergence Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelsConvergenceTheorem.html Subject classifications | ||||||||
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