Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.2 SUMMATION OF SERIES

4.2.1 Arithmetic series

Anarithmetic serieshas the characteristic that the difference between successive

terms is constant. The sum of a general arithmetic series is written

SN=a+(a+d)+(a+2d)+···+[a+(N−1)d]=

N∑− 1

n=0

(a+nd).

Rewriting the series in the opposite order and adding this term by term to the

original expression forSN, we find

SN=

N

2

[a+a+(N−1)d]=

N

2

(first term + last term). (4.2)

If an infinite number of such terms are added the series will increase (or decrease)

indefinitely; that is to say, it diverges.

Sum the integers between 1 and 1000 inclusive.

This is an arithmetic series witha=1,d=1andN= 1000. Therefore, using (4.2) we find

SN=

1000

2

(1 + 1000) = 500500,

which can be checked directly only with considerable effort.

4.2.2 Geometric series

Equation (4.1) is a particular example of ageometric series, which has the

characteristic that the ratio of successive terms is a constant (one-half in this

case). The sum of a geometric series is in general written

SN=a+ar+ar^2 +···+arN−^1 =

N∑− 1

n=0

arn,

whereais a constant andris the ratio of successive terms, thecommon ratio.The

sum may be evaluated by consideringSNandrSN:

SN=a+ar+ar^2 +ar^3 +···+arN−^1 ,

rSN=ar+ar^2 +ar^3 +ar^4 +···+arN.

If we now subtract the second equation from the first we obtain

(1−r)SN=a−arN,

and hence

SN=

a(1−rN)

1 −r

. (4.3)