Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Problem A1.13
Test for convergence the series
1 2
‡ 1 3 2
‡ 1 3 5 2
1 3 5 …2n 1†
‡‡ ‡:
2 2 4 2 4 6
1 3 5 …2n†
Hint: Neither the ratio test nor Raabe's test is applicable (show this). Try Gauss'
test.
Series of functions and uniform convergence
The series considered so far had the feature that u n depended just on n. Thus the
series, if convergent, is represented by just a number. We now consider series
whose terms are functions of x; u n ˆ u n …x†. There are many such series of functions.
The reader should be familiar with the power series in which the nth term is
a constant times x n :
S…x† ˆX1
nˆ0
a n x n :
…A1:4†
We can think of all previous cases as power series restricted to x ˆ 1. In later
sections we shall see Fourier series whose terms involve sines and cosines, and
other series in which the terms may be polynomials or other functions. In this
section we consider power series in x.
The convergence or divergence of a series of functions depends, in general, on
the values of x. With x in place, the partial sum Eq. (A1.2) now becomes a
function of the variable x:
s n …x† ˆu 1 ‡ u 2 …x†‡‡u n …x†:
as does the series sum. If we de®ne S…x† as the limit of the partial sum
S…x† ˆ lim s n …x† ˆX1
u n …x†;
n!1
nˆ0
…A1:5†
…A1:6†
then the series is said to be convergent in the interval [a, b] (that is, a x b), if
for each ">0 and each x in [a, b] we can ®nd N > 0 such that
jS…x†s n …x† j