Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 +···,