Power Series
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).
 |
and 
.
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Taylor series
