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Abel's limit theorem (Theorem)

Suppose that $ \sum a_{n} x^{n}$ has a radius of convergence $ r$ and that $ \sum a_{n} r^{n}$ is convergent. Then

$\displaystyle \lim_{x \to r^{-}} \sum a_{n} x^{n} = \sum a_{n} r^{n} = \sum (\lim_{x \to r^{-}} a_{n} x^{n})$



"Abel's limit theorem" is owned by vypertd.
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See Also: power series, Abel's multiplication rule for series


Attachments:
proof of Abel's limit theorem (Proof) by mathwizard
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Cross-references: convergent, radius of convergence
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This is version 2 of Abel's limit theorem, born on 2002-08-23, modified 2002-08-23.
Object id is 3331, canonical name is AbelsLimitTheorem.
Accessed 4309 times total.

Classification:
AMS MSC: 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
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