504 Chapter 8 • Infinite Series of Real Numbers
- Prove that if { ak} is square summable, then lim ak = 0.
k->oo - Prove that if I.: ak is a convergent nonnegative series then f V: con-
k= I
verges. - Prove that the sum of two square summable sequences is square summable.
8.6 Power Series
The study of power series has been an essential part of analysis for over 300
years, and remains so today. Since the early days of the development of calculus,
power series have been an indispensable tool for calculating the values of many
complicated algebraic and transcendental functions, and a powerful technique
for solving a wide variety of problems. Many brilliant discoveries in pure and
applied mathematics have been made using power series. Undergraduates are
often skeptical of such claims and only begin to accept the importance of power
series when they see them at work in other courses such as differential equations,
statistics, complex variables, and physics. In this section we present only the
essential core of this important subject. We begin by saying what we mean by
a power series.
Definition 8.6.1 A power series is a series of the form
00
:L>k(x -c)k,
k=Owhere c is a fixed real number and a ll the "coefficients" ak are real numbers.
Such a series is sometimes called a power series "in x - c" or a power series
"about c."
When c = 0, we have the power series
00
Lakxk,
k=O
which is said to be a power series "in x" or "about the origin."
If we regard power series as functions of x we can write
00 00
f(x) = L ak(x - c)k or f(x) = L akxk.
k=O k=OThen, of course, it is natural to ask what is the domain of such a function:
i.e., for what values of x does the given power series converge? Fortunately,
the set of points where a power series converges is a well-behaved set. In fact,