Page 280 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 271

THEOREMS ON UNIFORMLY CONVERGENT SERIES
If an infinite series of functions is uniformly convergent, it has many of the properties possessed by
sums of finite series of functions, as indicated in the following theorems.

Theorem 6. If fu n ðxÞg; n ¼ 1; 2; 3; ... are continuous in ½a; bŠ and if u n ðxÞ converges uniformly to the
sum SðxÞ in ½a; bŠ, then SðxÞ is continuous in ½a; bŠ.

Briefly, this states that a uniformly convergent series of continuous functions is a continuous
function. This result is often used to demonstrate that a given series is not uniformly convergent by
showing that the sum function SðxÞ is discontinuous at some point (see Problem 11.30).
In particular if x 0 is in ½a; bŠ, then the theorem states that

1 1 1
X X X
lim u n ðxÞ¼ lim u n ðxÞ¼ u n ðx 0 Þ
x!x 0 x!x 0
n¼1 n¼1 n¼1
where we use right- or left-hand limits in case x 0 is an endpoint of ½a; bŠ.
Theorem 7. If fu n ðxÞg; n ¼ 1; 2; 3; ... ; are continuous in ½a; bŠ and if u n ðxÞ converges uniformly to the
sum SðxÞ in ½a; bŠ, then

ð b 1 ð b
X
u n ðxÞ dx
SðxÞ dx ¼ ð4Þ
a n¼1 a
or
( )
ð 1 ð b
1
b X
X
u n ðxÞ dx
u n ðxÞ dx ¼ ð5Þ
a n¼1 n¼1 a
Briefly, a uniformly convergent series of continuous functions can be integrated term by term.
Theorem 8. If fu n ðxÞg; n ¼ 1; 2; 3; ... ; are continuous and have continuous derivatives in ½a; bŠ and if
0
u n ðxÞ converges to SðxÞ while u n ðxÞ is uniformly convergent in ½a; bŠ, then in ½a; bŠ
1
X
0 0
S ðxÞ¼ u n ðxÞ ð6Þ
n¼1
or
( )
d X X d
1
1
dx u n ðxÞ ¼ dx u n ðxÞ ð7Þ
n¼1 n¼1
This shows conditions under which a series can be differentiated term by term.
Theorems similar to the above can be formulated for sequences. For example, if fu n ðxÞg,
n ¼ 1; 2; 3; ... is uniformly convergent in ½a; bŠ, then

b b
ð ð
lim u n ðxÞ dx ¼ lim u n ðxÞ dx ð8Þ
n!1 a a n!1
which is the analog of Theorem 7.