Page 92 - Excel for Scientists and Engineers: Numerical Methods

Chapter 4

Number Series

Number series, such as

are important in many areas of mathematics, such as the evaluation of
transcendental functions, integrals or differential equations. Often, the sum of a
number series is used as an approximation to a function that can't be evaluated
directly. The approximation becomes more and more accurate as more terms are
added to the sum; for example, the value of e, the base of natural logarithms, can
be evaluated by means of the sum of an infinite series:

If the sum of a series approaches a finite value as the number of terms
approaches infinity, the series is said to be convergent. A series is divergent if
the sum approaches infinity (or does not converge to a definite value) when the
number of terms approaches infinity. Only convergent series will be discussed in
this chapter.
An alternating series in one in which the sign of each successive term is the
opposite of the preceding one. Such a series will always converge if the absolute
value of the nth term approaches zero.
Instead of a series of constant terms, a series may consist of variables, as
exemplified by the series

ao+a1x+a2x2+ a** +a&"+ ... (4-2)
A series of the form shown above, in which the terms are multiples of non-
negative integral powers of x, is called a power series.
Functions such as ex, sin x, cos x and others can be expressed in terms of the
sum of an infinite series. Of course, Excel already provides worksheet functions
to evaluate ex, sin x or cos x, but the ability to use number series in Excel
formulas increases the scope of calculations that you can perform.

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