Chapter 8
Infinite Series of Real
Numbers
Sections 8.1 and 8.2 cover basic concepts and the standard
convergence tests. Section 8.3 covers absolute and conditional
convergence, alternating series, and rearrangements. Section
8.4 explores Cauchy products of series. Section 8.5 explores
Abel's summation by parts, Dirichlet's test, and series of
products. Sections 8.6- 8.8 give a standard introduction to
power series and analytic functions. Uniform convergence is
left to Chapter 9.8.1 Basic Concepts and Examples
Definition 8.1.1 If {an} is a sequence of real numbers, the formal notation
(1)is called an infinite series, with nth term an. Corresponding to each infinite
series (1) there is a related sequence {Sn} called its sequence of partial sums,
(2)We say that an infinite series (1) converges to a real number S (or di-
verges) if the corresponding sequence of partial sums (2) converges to S (or
00
diverges). If lim Sn= S, we say that L an has sum S , and write
n~oo n=l
00
Lan= S. (3)
n=l
453