Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

The difference method may be easily extended to evaluate sums in which each

term can be expressed in the form

un=f(n)−f(n−m), (4.6)

wheremis an integer. By writing out the sum toNterms with each term expressed

in this form, and cancelling terms in pairs as before, we find

SN=

∑m

k=1

f(N−k+1)−

∑m

k=1

f(1−k).

Evaluate the sum

∑N

n=1

1

n(n+2)

.

Using partial fractions we find

un=−

[

1

2(n+2)

−

1

2 n

]

.

Henceun=f(n)−f(n−2) withf(n)=− 1 /[2(n+2)],andsothesumisgivenby

SN=f(N)+f(N−1)−f(0)−f(−1) =

3

4

−

1

2

(

1

N+2

+

1

N+1

)

.

In fact the difference method is quite flexible and may be used to evaluate

sums even when each term cannot be expressed as in (4.6). The method still relies,

however, on being able to writeunin terms of a single function such that most

terms in the sum cancel, leaving only a few terms at the beginning and the end.

This is best illustrated by an example.

Evaluate the sum

∑N

n=1

1

n(n+1)(n+2)

.

Using partial fractions we find

un=

1

2(n+2)

−

1

n+1

+

1

2 n

.

Henceun=f(n)− 2 f(n−1) +f(n−2) withf(n)=1/[2(n+ 2)]. If we write out the sum,
expressing each termunin this form, we find that most terms cancel and the sum is given
by

SN=f(N)−f(N−1)−f(0) +f(−1) =

1

4

+

1

2

(

1

N+2

−

1

N+1

)

.