MACLAURIN'S THEOREM:. 73 



If the function y n+1 (a; +0A) be such that by making n suffi- 



A n+1 

 ently great the term - - r/" 41 (# + #A) can 



as we please, then by carrying on the series 



ciently great the term - - r/" 41 (# + #A) can be made as small 



to as many terms as we please, we obtain a result differing as 

 little as we please from f (as + h) . Under these circumstances 

 then we may assert the truth of Taylor's Theorem. 



93. Taylor's Theorem is so called from its discoverer 

 Dr Brook Taylor; it was first published in 1715. The 

 theorem contained in equation (4) of Art. 92 is called 

 Lagrange's Theorem on the limits of Taylors Theorem. It 

 gives us an expression for the difference between f(x + h) 

 and the first n + 1 terms of its expansion by Taylor's Theorem, 

 or as it is called " the remainder after n+1 terms." 



94. To the expression f n+1 (as + 0h) which occurs in Art. 92, 

 we must assign the following meaning. "Let f(x) be dif- 

 ferentiated n + 1 times with respect to x, and in the final 

 result change x into x + 6h" We do not know anything.- of 

 0, except that it lies between and 1 ; it will generally be 

 a function of x and h, and hence, to differentiate f(x + 6K) 

 with respect to x, is not the same thing as to differentiate 

 f(x) with respect to x and then to change x into x + 6h. 



95. Maclauriris Theorem. 

 In the equation 



. + in^>*), 



put x = 0, we have then 



.... + *fia + -*Vw 



[ (714-1'' v ' 



