Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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.

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