Page 301 - Schaum's Outline of Theory and Problems of Advanced Calculus

292 INFINITE SERIES [CHAP. 11

Thus, R is the radius of convergence of na n x n 1 .
Note that the series of derivatives may or may not converge for values of x such that jxj¼ R.

1 n
X x
11.40. Illustrate Problem 11.39 by using the series .
2
n 3 n
n¼1

nþ1 2 2
n
x n jxj
lim u nþ1 n 3 ¼ lim
n!1 ðn þ 1Þ 3
¼ lim 2 nþ1 x n 2 jxj¼ 3
n!1 u n n!1 3ðn þ 1Þ
so that the series converges for jxj < 3. At x ¼ 3the series also converges, so that the interval of
convergence is 3 @ x @ 3.
The series of derivatives is
1 n 1 1 n 1
X nx X x
2
n 3 n ¼ n 3 n
n¼1 n¼1
By Problem 11.25(a)this has the interval of convergence 3 @ x < 3.
The two series have the same radius of convergence, i.e., R ¼ 3, although they do not have the same
interval of convergence.
Note that the result of Problem 11.39 can also be proved by the ratio test if this test is applicable. The
proof given there, however, applies even when the test is not applicable, as in the series of Problem 11.22.
11.41. Prove that in any interval within its interval of convergence a power series
ðaÞ represents a continuous function, say, f ðxÞ,
ðbÞ can be integrated term by term to yield the integral of f ðxÞ,
ðcÞ can be differentiated term by term to yield the derivative of f ðxÞ.
n
n
We consider the power series a n x ,although analogous results hold for a n ðx aÞ .
n
(a) This follows from Problem 11.33 and 11.34, and the fact that each term a n x of the series is continuous.
n
(b) This follows from Problems 11.33 and 11.35, and the fact that each term a n x of the series is continuous
and thus integrable.
(c) From Problem 11.39, the series of derivatives of a power series always converges within the interval of
convergence of the original power series and therefore is uniformly convergent within this interval.
Thus, the required result follows from Problems 11.33 and 11.36.
If a power series converges at one (or both) end points of the interval of convergence, it is possible to
establish (a) and (b)toinclude the end point (or end points). See Problem 11.42.

11.42. Prove Abel’s theroem that if a power series converges at an end point of its interval of conver-
gence, then the interval of uniform convergence includes this end point.
1
X k
For simplicity in the proof, we assume the power series to be a k x with the end point of its interval
k¼0
of convergence at x ¼ 1, so that the series surely converges for 0 @ x @ 1. Then we must show that the
series converges uniformly in this interval.
Let
n
R n ðxÞ¼ a n x þ a nþ1 x nþ1 þ a nþ2 x nþ2 þ ; R n ¼ a n þ a nþ1 þ a nþ2 þ
To prove the required result we must show that given any > 0, we can find N such that jR n ðxÞj < for
all n > N, where N is independent of the particular x in 0 @ x @ 1.