Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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

.