Page 274 - Schaum's Outline of Theory and Problems of Advanced Calculus

Infinite Series

The early developers of the calculus, including Newton and Leibniz, were well aware of the
importance of infinite series. The values of many functions such as sine and cosine were geometrically
obtainable only in special cases. Infinite series provided a way of developing extensive tables of values
for them.
This chapter begins with a statement of what is meant by infinite series, then the question of when
these sums can be assigned values is addressed. Much information can be obtained by exploring infinite
sums of constant terms; however, the eventual objective in analysis is to introduce series that depend on
variables. This presents the possibility of representing functions by series. Afterward, the question of
how continuity, differentiability, and integrability play a role can be examined.
The question of dividing a line segment into infinitesimal parts has stimulated the imaginations of
philosophers for a very long time. In a corruption of a paradox introduce by Zeno of Elea (in the fifth
century B.C.)a dimensionless frog sits on the end of a one-dimensional log of unit length. The frog
jumps halfway, and then halfway and halfway ad infinitum. The question is whether the frog ever
reaches the other end. Mathematically, an unending sum,
1 1 1
2 þ þ þ 2 n þ
4
is suggested. ‘‘Common sense’’ tells us that the sum must approach one even though that value is never
attained. We can form sequences of partial sums
1 1 1 1 1 1
S 1 ¼ ; S 2 ¼ þ ; ... ; S n ¼ þ þ þ n þ
2 2 4 2 4 2
and then examine the limit. This returns us to Chapter 2 and the modern manner of thinking about the
infinitesimal.
In this chapter consideration of such sums launches us on the road to the theory of infinite series.

DEFINITIONS OF INFINITE SERIES AND THEIR CONVERGENCE AND DIVERGENCE
Definition: The sum

1
X
S ¼ u n ¼ u 1 þ u 2 þ þ u n þ ð1Þ
n¼1
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