Maclaurin’s series

The form of f 0) can only be determined by experiment, or by means of atomistic hypotheses (cf. Chap. XVIII.). Since ce is known to change continuously and only comparatively slowly with rise of temperature, w e may expand J 0) in a Maclaurin s series (cf. H. M., 95) ... 

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... 

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. 

Concurrently with the above mathematical treatment there exists cinother approach to the problem of the determination of the expUcit form of the function t(AE) starting from equation (11), which is based on its MacLaurin s series expansion [57]... 

When a = 0, the series formed is known as Maclaurin s series ... 

Equation 7.9 can be expanded into MacLaurin s series, then truncated after the second term to give a simplified version, valid within the range — 1 < ( )ii/(t>m < 1 ... 

The inner and outer expansions for the dependent variables are related by expanding the latter in Maclaurin s series in n, expressing these in terms of the inner variable 5, and comparing the resulting series with the inner variable expansions at each order in e. This results in the following ... 

A Maclaurin s series expansion in density around p = 0 yields ... 

---