Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

4.4 OPERATIONS WITH SERIES


is always less thanuNfor allmandun→0asn→∞, the alternating series


converges. It is clear that an analogous proof can be constructed in the case


whereNis even.


Determine whether the following series converges:
∑∞

n=1

(−1)n+1

1


n

=1−


1


2


+


1


3


−···.


This alternating series clearly satisfies conditions (i) and (ii) above and hence converges.
However, as shown above by the method of grouping terms, the corresponding series with
all positive terms is divergent.


4.4 Operations with series

Simple operations with series are fairly intuitive, and we discuss them here only


for completeness. The following points apply to both finite and infinite series


unless otherwise stated.


(i) If


un=Sthen


kun=kSwherekis any constant.
(ii) If


un=Sand


vn=Tthen


(un+vn)=S+T.
(iii) If


un=Sthena+


un=a+S. A simple extension of this trivial result
shows that the removal or insertion of a finite number of terms anywhere
in a series does not affect its convergence.
(iv) If the infinite series


unand


vnare both absolutely convergent then
the series


wn,where

wn=u 1 vn+u 2 vn− 1 +···+unv 1 ,

is also absolutely convergent. The series


wnis called theCauchy product
of the two original series. Furthermore, if


unconverges to the sumS
and


vnconverges to the sumTthen


wnconverges to the sumST.
(v) It is not true in general that term-by-term differentiation or integration of
a series will result in a new series with the same convergence properties.

4.5 Power series

A power series has the form


P(x)=a 0 +a 1 x+a 2 x^2 +a 3 x^3 +···,

wherea 0 ,a 1 ,a 2 ,a 3 etc. are constants. Such series regularly occur in physics and


engineering and are useful because, for|x|<1, the later terms in the series may


become very small and be discarded. For example the series


P(x)=1+x+x^2 +x^3 +···,
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