Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

4.2 SUMMATION OF SERIES


Sum the series

S=2+

5


2


+


8


22


+


11


23


+···.


This is an infinite arithmetico-geometric series witha=2,d=3andr=1/2. Therefore,
from (4.5), we obtainS= 10.


4.2.4 The difference method

The difference method is sometimes useful in summing series that are more


complicated than the examples discussed above. Let us consider the general series


∑N

n=1

un=u 1 +u 2 +···+uN.

If the terms of the series,un, can be expressed in the form


un=f(n)−f(n−1)

for some functionf(n) then its (partial) sum is given by


SN=

∑N

n=1

un=f(N)−f(0).

This can be shown as follows. The sum is given by

SN=u 1 +u 2 +···+uN

and sinceun=f(n)−f(n−1), it may be rewritten


SN=[f(1)−f(0)] + [f(2)−f(1)] +···+[f(N)−f(N−1)].

By cancelling terms we see that


SN=f(N)−f(0).

Evaluate the sum
∑N

n=1

1


n(n+1)

.


Using partial fractions we find


un=−

(


1


n+1


1


n

)


.


Henceun=f(n)−f(n−1) withf(n)=− 1 /(n+1), and so the sumis given by


SN=f(N)−f(0) =−

1


N+1


+1=


N


N+1


.

Free download pdf