Principles of Mathematics in Operations Research

13

13 Power Series and Special Functions

In mathematics, power series are devices that make it possible to employ

much of the analytical machinery in settings that do not have natural notions

of "convergence". They are also useful, especially in combinatorics, for provid-

ing compact representations of sequences and for finding closed formulas for

recursively defined sequences, known as the method of generating functions.

We will discuss first the notion of series, succeeded by operations on series

and tests for convergence/divergence. After power series is formally defined,

we will generate exponential, logarithmic and trigonometric functions in this

chapter. Fourier series, gamma and beta functions will be discussed as well.

13.1 Series

13.1.1 Notion of Series

Definition 13.1.1 An expression

oo oo

X/

Uk =

X/

Uk

~

u

°

+Ui+

"

2

^—

fc=0 0

where the numbers Uk (terms of the series) depend on the index k — 0,1,2,...

is called a (number) series. The number

Sn — u 0 + ui -i \-un, n = 0,1,...

is called the nth partial sum of the above series.
We say that the series is convergent if the limit, limn-^ Sn = S, exists.
In this case, we write

oo

S = Up + Ui + U2 + • • • = }j Uk

k=0

and call S the sum of the series; we also say that the series converges to S.