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 «th term approaches zero.
Instead of a series of constant terms, a series may consist of variables, as exemplified by the series
A series of the form shown above, in which the terms are multiples of nonnegative 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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