Page 281 - Schaum's Outline of Theory and Problems of Advanced Calculus

272 INFINITE SERIES [CHAP. 11

POWER SERIES
A series having the form
1
2 X n
a 0 þ a 1 x þ a 2 x þ ¼ a n x ð9Þ
n¼0
where a 0 ; a 1 ; a 2 ; ... are constants, is called a power series in x.Itis often convenient to abbreviate the
n
series (9)as a n x .
In general a power series converges for jxj < R and diverges for jxj > R, where the constant R is
called the radius of convergence of the series. For jxj¼ R, the series may or may not converge.
The interval jxj < R or R < x < R, with possible inclusion of endpoints, is called the interval of
convergence of the series. Although the ratio test is often successful in obtaining this interval, it may fail
and in such cases, other tests may be used (see Problem 11.22).
The two special cases R ¼ 0and R ¼1 can arise. In the first case the series converges only for
x ¼ 0; in the second case it converges for all x, sometimes written 1 < x < 1 (see Problem 11.25).
When we speak of a convergent power series, we shall assume, unless otherwise indicated, that R > 0.
Similar remarks hold for a power series of the form (9), where x is replaced by ðx aÞ.

THEOREMS ON POWER SERIES
Theorem 9. A power series converges uniformly and absolutely in any interval which lies entirely within
its interval of convergence.

Theorem 10. A power series can be differentiated or integrated term by term over any interval lying
entirely within the interval of convergence. Also, the sum of a convergent power series is continuous in
any interval lying entirely within its interval of convergence.
This follows at once from Theorem 9 and the theorems on uniformly convergent series on Pages 270
and 271. The results can be extended to include end points of the interval of convergence by the
following theorems.
Theorem 11. Abel’s theorem. When a power series converges up to and including an endpoint of its
interval of convergence, the interval of uniform convergence also extends so far as to include this
endpoint. See Problem 11.42.
1
X n
Theorem 12. Abel’s limit theorem. If a n x converges at x ¼ x 0 , which may be an interior point or an
n¼0
endpoint of the interval of convergence, then
( )

1 1 1
X X X
lim a n x n ¼ lim a n x n ¼ a n x 0 n ð10Þ
x!x 0 x!x 0
n¼0 n¼0 n¼0
If x 0 is an end point, we must use x ! x 0 þ or x ! x 0 in (10) according as x 0 is a left- or right-hand
end point.
This follows at once from Theorem 11 and Theorem 6 on the continuity of sums of uniformly
convergent series.

OPERATIONS WITH POWER SERIES
In the following theorems we assume that all power series are convergent in some interval.
Theorem 13. Two power series can be added or subtracted term by term for each value of x common to
their intervals of convergence.