Abel's limit theoremAbelsLimitTheorem2002-08-23 09:13:162002-08-23 09:14:58Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereSuppose that $\sum a_{n} x^{n}$ has a radius of convergence $r$ and that $\sum a_{n} r^{n}$ is convergent. Then
$$\lim_{x \to r^{-}} \sum a_{n} x^{n} = \sum a_{n} r^{n} = \sum (\lim_{x \to r^{-}} a_{n} x^{n})$$