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).
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. Monthly102, 698-705, 1995.Niven, I. "Formal
Power Series." Amer. Math. Monthly76, 871-889, 1969.Pólya,
G. Mathematics
and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference. Princeton,
NJ: Princeton University Press, 1990.
is an infinite sum of
the form
are integers, real numbers,
complex numbers, or any other quantities of a given
type.
.
.
in some finite open interval 
and diverges if 
or 
. At the points 
and 
, the series may converge absolutely, converge conditionally,
or diverge.
. 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).
and 
.
