Page 279 - Schaum's Outline of Theory and Problems of Advanced Calculus

270 INFINITE SERIES [CHAP. 11

n
EXAMPLE 2. If u n ¼ x and 0 @ x @ 1, the sequence is not uniformly convergent because (think of the function
FðxÞ¼ 0, 0 @ x < 1, Fð1Þ¼ 1Þ
n
n
jx 0j < when x < ;
thus
n ln x < ln :
On the interval 0 @ x < 1, and for 0 < < 1, both members
ln
of the inequality are negative, therefore, n > : Since
ln ln 1 ln ln x
,it follows that we must choose N
lnð= Þ
ln x ¼ ln 1 nn x ¼ lnð1=xÞ
such that
ln 1=
n > N >
ln 1=x
1
From this expression we see that ! 0 then ln !1 and
1
also as x ! 1 from the left ln ! 0 from the right; thus, in either
x
case, N must increase without bound. This dependency on both Fig. 11-1
and x demonstrations that the sequence is not uniformly
convergent. For a pictorial view of this example, see Fig. 11-1.

SPECIAL TESTS FOR UNIFORM CONVERGENCE OF SERIES
1. Weierstrass M test. If sequence of positive constants M 1 ; M 2 ; M 3 ; ... can be found such that
in some interval
(a) ju n ðxÞj @ M n n ¼ 1; 2; 3; ...
(b) M n converges
then u n ðxÞ is uniformly and absolutely convergent in the interval.

1
X cos nx cos nx 1 X 1
EXAMPLE. is uniformly and absolutely convergent in ½0; 2 Š since @ and
n 2 n 2 n 2 n 2
n¼1
converges.
This test supplies a sufficient but not a necessary condition for uniform convergence, i.e., a
series may be uniformly convergent even when the test cannot be made to apply.
One may be led because of this test to believe that uniformly convergent series must be
absolutely convergent, and conversely. However, the two properties are independent, i.e., a
series can be uniformly convergent without being absolutely convergent, and conversely. See
Problems 11.30, 11.127.
2. Dirichlet’s test. Suppose that
(a) the sequence fa n g is a monotonic decreasing sequence of positive constants having limit
zero,
(b) there exists a constant P such that for a @ x @ b

ju 1 ðxÞþ u 2 ðxÞþ þ u n ðxÞj < P for all n > N:
Then the series
1
X
a 1 u 1 ðxÞþ a 2 u 2 ðxÞþ ¼ a n u n ðxÞ
n¼1
is uniformly convergent in a @ x @ b.