Page 275 - Schaum's Outline of Theory and Problems of Advanced Calculus

266 INFINITE SERIES [CHAP. 11

is an infinite series. Its value, if one exists, is the limit of the sequence of partial sums fS n g
S ¼ lim S n ð2Þ
n!1
If there is a unique value, the series is said to converge to that sum, S.If there is not a unique sum
the series is said to diverge.
1
1
X
Sometimes the character of a series is obvious. For example, the series n generated by the
2
n¼1
1
X
frog on the log surely converges, while n is divergent. On the other hand, the variable series
n¼1
2 3 4 5
1 x þ x x þ x x þ
raises questions. 1
This series may be obtained by carrying out the division .If 1 < x < 1, the sums S n yields an
1 1 x
approximations to and (2)is the exact value. The indecision arises for x ¼ 1. Some very great
1 x
1
mathematicians, including Leonard Euler, thought that S should be equal to ,asis obtained by
2
1
substituting 1 into . The problem with this conclusion arises with examination of
1 x
1 1 þ 1 1þ 1 1 þ and observation that appropriate associations can produce values of 1 or
0. Imposition of the condition of uniqueness for convergence put this series in the category of divergent
and eliminated such possibility of ambiguity in other cases.
FUNDAMENTAL FACTS CONCERNING INFINITE SERIES
1. If u n converges, then lim u n ¼ 0 (see Problem 2.26, Chap. 2). The converse, however, is not
n!1
necessarily true, i.e., if lim u n ¼ 0, u n may or may not converge. It follows that if the nth
n!1
term of a series does not approach zero the series is divergent.
2. Multiplication of each term of a series by a constant different from zero does not affect the
convergence or divergence.
3. Removal (or addition) of a finite number of terms from (or to) a series does not affect the
convergence or divergence.

SPECIAL SERIES
1
X n 1 2
1. Geometric series ar ¼ a þ ar þ ar þ , where a and r are constants, converges to
n
a n¼1 að1 r Þ
if jrj < 1 and diverges if jrj A 1.
S ¼ The sum of the first n terms is S n ¼
1 r 1 r
(see Problem 2.25, Chap. 2).
1
X 1 1 1 1
2. The p series ¼ þ þ þ ; where p is a constant, converges for p > 1and diverges
n p 1 p 2 p 3 p
n¼1
for p @ 1. The series with p ¼ 1is called the harmonic series.
TESTS FOR CONVERGENCE AND DIVERGENCE OF SERIES OF CONSTANTS
More often than not, exact values of infinite series cannot be obtained. Thus, the search turns
toward information about the series. In particular, its convergence or divergence comes in question.
The following tests aid in discovering this information.