Page 276 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 11] INFINITE SERIES 267

1. Comparison test for series of non-negative terms.
(a) Convergence. Let v n A 0 for all n > N and suppose that v n converges. Then if
0 @ u n @ v n for all n > N, u n also converges. Note that n > N means from some
term onward. Often, N ¼ 1.
1 1 X 1 X 1
EXAMPLE: Since @ and converges, also converges.
n
n
2 þ 1 2 n 2 n 2 þ 1
(b) Divergence. Let v n A 0 for all n > N and suppose that v n diverges. Then if u n A v n for
all n > N, u n also diverges.
1 1 X 1 X 1
1
1
EXAMPLE: Since > and diverges, also diverges.
ln n n n ln n
n¼2 n¼2
2. The Limit-Comparison or Quotient Test for series of non-negative terms.
u n
(a)If u n A 0 and v n A 0and if lim ¼ A 6¼ 0or 1, then u n and v n either both converge
or both diverge. n!1 v n
(b)If A ¼ 0in(a)and v n converges, then u n converges.
(c)If A ¼1 in (a) and v n diverges, then u n diverges.
This test is related to the comparison test and is often a very useful alternative to it. In
p
particlar, taking v n ¼ 1=n ,we have from known facts about the p series the
p
Theorem 1. Let lim n u n ¼ A. Then
n!1
(i) u n converges if p > 1 and A is finite.
(ii) u n diverges if p @ 1and A 6¼ 0(A may be infinite).

X n 2 n 1
EXAMPLES: 1: converges since lim n ¼ :
3
3
4n 2 n!1 4n 2 4
ln n ln n
X 1=2
2: p ffiffiffiffiffiffiffiffiffiffiffi diverges since lim n ¼1:
n þ 1 n!1 1=2
ðn þ 1Þ
3. Integral test for series of non-negative terms.
If f ðxÞ is positive, continuous, and monotonic decreasing for x A N and is such that
f ðnÞ¼ u n ; n ¼ N; N þ 1; N þ 2; ... , then u n converges or diverges according as
ð ð M
1
f ðxÞ dx ¼ lim f ðxÞ dx converges or diverges. In particular we may have N ¼ 1, as
N M!1 n
is often true in practice.
This theorem borrows from the next chapter since the integral has an unbounded upper
limit. (It is an improper integral. The convergence or divergence of these integrals is defined in
much the same way as for infinite series.)
1 ð M
X 1 dx 1
EXAMPLE: converges since lim ¼ lim 1 exists.
n 2 1 x 2 M
n¼1 M!1 M!1
4. Alternating series test. An alternating series is one whose successive terms are alternately
positive and negative.
An alternating series converges if the following two conditions are satisfied (see Problem
11.15).
for n A N (Since a fixed number of terms does not affect the conver-
(a) ju nþ1 j @ ju n j
gence or divergence of a series, N may be any positive integer. Frequently it is chosen to
be 1.)

(b) lim u n ¼ 0or lim ju n j¼ 0
n!1 n!1