Page 278 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 269

THEOREMS ON ABSOLUTELY CONVERGENT SERIES
Theorem 4. (Rearrangement of Terms) The terms of an absolutely convergent series can be rearranged
in any order, and all such rearranged series will converge to the same sum. However, if the terms of a
conditionally convergent series are suitably rearranged, the resulting series may diverge or converge to
any desired sum (see Problem 11.80).
Theorem 5. (Sums, Differences, and Products) The sum, difference, and product of two absolutely
convergent series is absolutely convergent. The operations can be performed as for finite series.

INFINITE SEQUENCES AND SERIES OF FUNCTIONS, UNIFORM CONVERGENCE
We opened this chapter with the thought that functions could be expressed in series form. Such
representation is illustrated by
x 3 x 5 n 1 x 2n 1
3! 5! ð2n 1Þ!
sin x ¼ x þ þ þð 1Þ þ
where
3 n 2k 1
x X x
sin x ¼ lim S n ; with S 1 ¼ x; S 2 ¼ x ; ... S n ¼ ð 1Þ k 1 :
3! ð2k 1Þ!
n!1
k¼1
Observe that until this section the sequences and series depended on one element, n. Now there is
variation with respect to x as well. This complexity requires the introduction of a new concept called
uniform convergence, which, in turn, is fundamental in exploring the continuity, differentiation, and
integrability of series.
Let fu n ðxÞg; n ¼ 1; 2; 3; .. . be a sequence of functions defined in ½a; bŠ. The sequence is said to
converge to FðxÞ,orto have the limit FðxÞ in ½a; bŠ,if for each > 0and each x in ½a; bŠ we can find
N > 0 such that ju n ðxÞ FðxÞj < for all n > N.In such case we write lim u n ðxÞ¼ FðxÞ. The number
n!1
N may depend on x as well as .Ifitdepends only on and not on x, the sequence is said to converge to
FðxÞ uniformly in ½a; bŠ or to be uniformly convergent in ½a; bŠ.
The infinite series of functions

1
X
u n ðxÞ¼ u 1 ðxÞþ u 2 ðxÞþ u 3 ðxÞþ ð3Þ
n¼1
is said to be convergent in ½a; bŠ if the sequence of partial sums fS n ðxÞg, n ¼ 1; 2; 3; ... ; where
S n ðxÞ¼ u 1 ðxÞþ u 2 ðxÞþ þ u n ðxÞ,is convergent in ½a; bŠ. In such case we write lim S n ðxÞ¼ SðxÞ
and call SðxÞ the sum of the series. n!1
It follows that u n ðxÞ converges to SðxÞ in ½a; bŠ if for each > 0 and each x in ½a; bŠ we can find
N > 0 such that jS n ðxÞ SðxÞj < for all n > N.If N depends only on and not on x, the series is called
uniformly convergent in ½a; bŠ.
Since SðxÞ S n ðxÞ¼ R n ðxÞ, the remainder after n terms, we can equivalently say that u n ðxÞ is
uniformly convergent in ½a; bŠ if for each > 0we can find N depending on but not on x such that
jR n ðxÞj < for all n > N and all x in ½a; bŠ.
These definitions can be modified to include other intervals besides a @ x @ b, such as a < x < b,
and so on.
The domain of convergence (absolute or uniform) of a series is the set of values of x for which the
series of functions converges (absolutely or uniformly).

n
1
EXAMPLE 1. Suppose u n ¼ x =n and @ x @ 1. Now think of the constant function FðxÞ¼ 0onthisinterval.
2
n
For any > 0 and any x in the interval, there is N such that for all n > Nju n FðxÞj < , i.e., jx =nj < .Since the
limit does not depend on x,the sequence is uniformly convergent.