PlanetMath: identity theorem of power series

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identity theorem of power series

(Theorem)

If the radii of convergence of the power series $\sum_{n=0}^\infty a_n(z-z_0)^n$ and $\sum_{n=0}^\infty b_n(z-z_0)^n$ are positive and the sums of the series are equal in infinitely many points which have as an accumulation point, then the both series are identical, i.e. for each $n = 0,\,1,\,2,\,\ldots$

"identity theorem of power series" is owned by pahio.

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Attachments:: proof of identity theorem of power series (Proof) by rspuzio
indirect proof of identity theorem of power series (Proof) by pahio
proof of identity theorem of power series (Proof) by rspuzio

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Cross-references: accumulation point, points, series, sums, positive, power series
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This is version 1 of identity theorem of power series, born on 2007-03-04.
Object id is 9017, canonical name is IdentityTheoremOfPowerSeries.
Accessed 1169 times total.

Classification:

AMS MSC:	30B10 (Functions of a complex variable :: Series expansions :: Power series )
	40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)

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