Maclaurin 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,... 

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... 

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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