Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

some sort of relationship between successive terms. For example, if thenth term

of a series is given by

un=

1

2 n

,

forn=1, 2 , 3 ,...,Nthen the sum of the firstNterms will be

SN=

∑N

n=1

un=

1

2

+

1

4

+

1

8

+···+

1

2 N

. (4.1)

It is clear that the sum of a finite number of terms is always finite, provided

that each term is itself finite. It is often of practical interest, however, to consider

the sum of a series with an infinite number of finite terms. The sum of an

infinite number of terms is best defined by first considering the partial sum

of the firstNterms,SN. If the value of the partial sumSNtends to a finite

limit,S,asNtends to infinity, then the series is said to converge and its sum

is given by the limitS. In other words, the sum of an infinite series is given

by

S= lim

N→∞

SN,

provided the limit exists. For complex infinite series, ifSNapproaches a limit

S=X+iYasN→∞, this means thatXN→XandYN→Yseparately, i.e.

the real and imaginary parts of the series are each convergent series with sums

XandYrespectively.

However, not all infinite series have finite sums. AsN→∞, the value of the

partial sumSNmay diverge: it may approach +∞or−∞, or oscillate finitely

or infinitely. Moreover, for a series where each term depends on some variable,

its convergence can depend on the value assumed by the variable. Whether an

infinite series converges, diverges or oscillates has important implications when

describing physical systems. Methods for determining whether a series converges

are discussed in section 4.3.

4.2 Summation of series

It is often necessary to find the sum of a finite series or a convergent infinite

series. We now describe arithmetic, geometric and arithmetico-geometric series,

which are particularly common and for which the sums are easily found. Other

methods that can sometimes be used to sum more complicated series are discussed

below.