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Abel's Limit Theorem

  • Definition: Assume that we have f(x) = sum[(a_n)x^n] as n goes from 0 to infinity, and if -r < x < r. If the series also converges at x = r, then the limit as x -> r- of f(x) exists and we have lim f(x) as x -> r- = sum[(a_n)r^n] as n goes from 0 to infinity.

    Source: Mathematical Analysis, second edition by Tom M. Apostol