Power Series

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A power series in a variablezis an infinite总和 of the form

 sum_(i=0)^inftya_iz^i,

哪里a_iintegersreal numberscomplex numbers, or any other quantities of a given type.

Pólya conjectured that if a功能has a power series withinteger coefficientsradius of convergence1, then either the功能理性的or theunit circleis anatural boundary(Pólya 1990, pp. 43 and 46). This conjecture was stated by G. Polya in 1916 and proved to be correct by Carlson (1921) in a result that is now regarded as a classic of early 20th centurycomplex analysis

For any power series, one of the following is true:

1. The series converges only forz=0

2. The series converges absolutely for allz

三.The series converges absolutely for allzin some finite open interval(-R,R)and diverges ifz<-Rz>RAt the pointsz=Rz=-R, the series may converge absolutely, converge conditionally, or diverge.

To determine the interval of convergence, apply theratio testabsolute convergenceand solve forzA power series may be differentiated or integrated within the interval of convergence. Convergent power series may be multiplied and divided (if there is no division by zero).

 sum_(k=1)^inftyk^(-p)

converges如果p>1diverges如果0<p<=1

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