Page 277 - Schaum's Outline of Theory and Problems of Advanced Calculus

268 INFINITE SERIES [CHAP. 11

n 1 n 1
1
X 1
EXAMPLE. 1 1 1 1 ð 1Þ ð 1Þ , ju n j¼ ,
2 3 4 5 n ,we have u n ¼ n n
For the series 1 þ þ ¼
1 n¼1
. Then for n A 1, ju nþ1 j @ ju n j. Also lim ju n j¼ 0. Hence, the series converges.
ju nþ1 j¼
n þ 1 n!1
Theorem 2. The numerical error made in stopping at any particular term of a convergent alternating
series which satisfies conditions (a) and (b)is less than the absolute value of the next term.
1
1
1
1
EXAMPLE. If we stop at the 4th term of the series 1 þ þ ,the error made is less than
3
2
4
5
1 ¼ 0:2.
5
5. Absolute and conditional convergence. The series u n is called absolutely convergent if ju n j
converges. If u n converges but ju n j diverges, then u n is called conditionally convergent.
Theorem 3. If ju n j converges, then u n converges. In words, an absolutely convergent series is
convergent (see Problem 11.17).
1 1 1 1 1 1
EXAMPLE 1. þ þ þ is absolutely convergent and thus convergent, since the
1 2 2 2 3 2 4 2 5 2 6 2
1 1 1 1
series of absolute values þ þ þ þ converges.
1 2 2 2 3 2 4 2
1 1 1 1 1 1 1 1 1
EXAMPLE 2. 1 þ þ converges, but 1 þ þ þ þ diverges. Thus, 1 þ þ
2 3 4 2 3 4 2 3 4
is conditionally convergent.
Any of the tests used for series with non-negative terms can be used to test for absolute
convergence. Also, tests that compare successive terms are common. Tests 6, 8, and 9 are of
this type.

u nþ1
6. Ratio test. Let lim ¼ L. Then the series u n

n!1 u n
(a) converges (absolutely) if L < 1
(b) diverges if L > 1.
If L ¼ 1 the test fails.
p ffiffiffiffiffiffiffiffi
7. The nth root test. Let lim n ju n j ¼ L. Then the series u n
n!1
(a) converges (absolutely) if L < 1
(b) diverges if L > 1:
If L ¼ 1 the test fails.

u n þ 1
8. Raabe’s test. Let lim n 1 ¼ L. Then the series u n

u n
n!1
(a) converges (absolutely) if L > 1
(b) diverges or converges conditionally if L < 1.
If L ¼ 1 the test fails.
This test is often used when the ratio tests fails.

u nþ1 L c n
9. Gauss’ test. If ¼ 1 þ , where jc n j < P for all n > N, then the series u n
n n 2
u n
(a) converges (absolutely) if L > 1
(b) diverges or converges conditionally if L @ 1.
This test is often used when Raabe’s test fails.