A power series in a variable  is an infinite
sum of
the form
where  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  .
2. The series converges absolutely for all  .
3. The series converges absolutely for all  in some finite
open interval  and diverges if  or  . At the points  and  , 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  . 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).
converges if  and diverges if  .
Portions of this entry contributed by Folkmar
Bornemann
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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