4

Series and limits

4.1 Series

Many examples exist in the physical sciences of situations where we are presented

with asum of termsto evaluate. For example, we may wish to add the contributions

from successive slits in a diffraction grating to find the total light intensity at a

particular point behind the grating.

A series may have either a finite or infinite number of terms. In either case, the

sum of the firstNterms of a series (often called a partial sum) is written

SN=u 1 +u 2 +u 3 +···+uN,

where the terms of the seriesun,n=1, 2 , 3 ,...,Nare numbers, that may in

general be complex. If the terms are complex thenSNwill in general be complex

also, and we can writeSN=XN+iYN,whereXNandYNare the partial sums of

the real and imaginary parts of each term separately and are therefore real. If a

series has onlyNterms then the partial sumSNis of course the sum of the series.

Sometimes we may encounter series where each term depends on some variable,

x, say. In this case the partial sum of the series will depend on the value assumed

byx. For example, consider the infinite series

S(x)=1+x+

x^2

2!

+

x^3

3!

+···.

This is an example of a power series; these are discussed in more detail in

section 4.5. It is in fact the Maclaurin expansion of expx(see subsection 4.6.3).

ThereforeS(x)=expxand, of course, varies according to the value of the

variablex. A series might just as easily depend on a complex variablez.

A general, random sequence of numbers can be described as a series and a sum

of the terms found. However, for cases of practical interest, there will usually be