Series

4
Series and limits
4.1 Series
Many examples exist in the physical sciences of situations where we are presented
with a sum of terms to 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 first N terms of a series (often called a partial sum) is written
SN = u1 + u2 + u3 + · · · + uN ,
where the terms of the series un , n = 1, 2, 3, . . . , N are numbers, that may in
general be complex. If the terms are complex then SN will in general be complex
also, and we can write SN = XN + iYN , where XN and YN are the partial sums of
the real and imaginary parts of each term separately and are therefore real. If a
series has only N terms then the partial sum SN is 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
by x. For example, consider the infinite series
S(x) = 1 + x +
x3
x2
+
+ ··· .
2!
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 exp x (see subsection 4.6.3).
Therefore S(x) = exp x and, of course, varies according to the value of the
variable x. A series might just as easily depend on a complex variable z.
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
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