Maclaurins theorem

Maclaurin s theorem determines the law for the expansion of a function of a single variable in a series of ascending powers of that variable. Let the variable be denoted by x, then, 

Assume that f(x) can be developed in ascending powers of xf like the series used in the preceding section, namely, 

There are several methods for the development of functions in series, depending on algebraic, trigonometrical, or other processes. The one of greatest utility is known as Taylor s theorem. Mac-laurin s1 theorem is but a special case of Taylor s. We shall work from the special to the general. 

1 The came is here a historical misnomer. Taylor published his series in 1715. In 1717, Stirling showed that the series under consideration was a special case of Taylor s. Twenty-five years after this Maclaurin independently published Stirling s series. But then both Maclaurin and Stirling, adds De Morgan, would have been astonished to know that a particular case of Taylor s theorem would be called by either of their names 

The series on the right-hand side is known as Maclaurin s Series. The first term is what the series becomes when x = 0 the second term is what the first derivative of the function becomes when x = 0, multiplied by x the third term is the product of the second derivative of the function when x = 0, into x2 divided by factorial 2... 

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In order to expand any function by Maclaurin s theorem, the successive differential coefficients of u are to be computed and x then equated to zero. This fixes the values of the different constants. 

The development by Maclaurin s series cannot be used if the function or any of its derivatives becomes infinite or discontinuous when x is equated to zero. For example, the first differential coefficient of f(x) = >Jx, is x which is infinite for x = 0, in other words, the series is no longer convergent. The same thing will be found with the functions log a , cot a , 1/x, a1,x and sec lx. Some of these functions may, however, be developed as a fractional or some other simple function of x, or we may use Taylor s theorem. 

While Maclaurin s theorem evaluates the series upon the assumption that the variable becomes zero, Taylor s theorem deduces a value for the series when x = a. Let z = a, then y = 0, and we get... 

Maclaurin s and Taylor s series are slightly different expressions for the same thing. The one form can be converted into the other by substituting f x + y) for f(x) in Maclaurin s theorem, or by putting y = 0 in Taylor s. 

The numerical tables of the trigonometrical functions are calculated by means of Taylor s or by Maclaurin s theorems. For example, by Maclaurin s theorem. 

Just as Maclaurin s theorem is a special case of Taylor s, so the latter is a special form of the more general Lagrange s theorem, and the latter, in turn, a special form of Laplace s theorem. There is no need for me to enter into extended details, but I shall have something to say about Lagrange s theorem. 

For cases in which a is to be made equal to zero, the numerator and denominator may be expanded at onoe by Maclaurin s theorem without any preliminary substitution for x. For instance, the trigonometrical function (sin x)/x approaches unity when x converges towards zero. This is seen directly. Develop sin x in ascending powers of x by Taylor s or Maclaurin s theorems. We thus obtain... 

This result is known as Taylor s theorem, and the expansion is a Taylor series. The case xq = 0, given by Eq. (7.40), is sometimes called a Maclaurin series. 

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