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Power Series


A power series in a variable z is an infinite sum of the form

 sum_(i=0)^inftya_iz^i,

where a_i are integers, real numbers, complex numbers, or any other quantities of a given type.

Pólya conjectured that if a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural 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 century complex analysis.

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

1. The series converges only for z=0.

2. The series converges absolutely for all z.

3. The series converges absolutely for all z in some finite open interval (-R,R) and diverges if z<-R or z>R. At the points z=R and z=-R, the series may converge absolutely, converge conditionally, or diverge.

To determine the interval of convergence, apply the ratio test for absolute convergence and solve for z. A 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 if p>1 and diverges if 0<p<=1.


See also

Binomial Series, Convergence Tests, Formal Power Series, Generating Function, Laurent Series, Maclaurin Series, Multinomial Series, p-Series, Polynomial, Power Set, Quotient-Difference Algorithm, Radius of Convergence, Recursive Sequence, Series, Series Reversion, Taylor Series Explore this topic in the MathWorld classroom

Portions of this entry contributed by Folkmar Bornemann

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References

Arfken, G. "Power Series." §5.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 313-321, 1985.Carlson, F. "Über Potenzreihen mit ganzzahligen Koeffizienten." Math. Z. 9, 1-13, 1921.Hanrot, G.; Quercia, M.; and Zimmermann, P. "Speeding Up the Division and Square Root of Power Series." Report RR-3973. INRIA, Jul 2000. http://www.inria.fr/rrrt/rr-3973.html.Myerson, G. and van der Poorten, A. J. "Some Problems Concerning Recurrence Sequences." Amer. Math. Monthly 102, 698-705, 1995.Niven, I. "Formal Power Series." Amer. Math. Monthly 76, 871-889, 1969.Pólya, G. Mathematics and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference. Princeton, NJ: Princeton University Press, 1990.

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Power Series

Cite this as:

Bornemann, Folkmar and Weisstein, Eric W. "Power Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PowerSeries.html

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