Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/254






 * align="right" | (a)
 * align="left" | $$(x+h)^n = x^n + nx^{n-1}h + \tfrac{n(n-1)}{2!}x^{n-2}h^2 + \tfrac{n(n-1)(n-2)}{3!}x^{n-3}h^3 + \cdots.$$
 * align="right" | (b)
 * align="left" | $$e^{x+h} = e^x \left( 1 + h + \tfrac{h^2}{2!} + \tfrac{h^3}{3!} + \cdots\right).$$
 * }
 * align="left" | $$e^{x+h} = e^x \left( 1 + h + \tfrac{h^2}{2!} + \tfrac{h^3}{3!} + \cdots\right).$$
 * }

A particular case of Taylor's Theorem is found by placing $$a=0$$ in  (61), &sect;144,  giving

where $$x_1$$ lies between 0 and $$x$$. (64) is called Maclaurin's Theorem. The right-hand member is evidently a series in $$x$$ in the same sense that (62), &sect;144, is a series in $$x - a$$.

Placing $$a = 0$$ in (62), &sect;144 we get Maclaurin's Series' ,