XXll USEFUL FORMULAS. 



6. Taylor's and Maclaurin's Series. 



a. Taylor's series. 



If //■ =y(.v -[- //), any finite and continuous function of x -\- /i, h being an 

 arbitrary increment to x; and if dicjdx, d'^uldx'^, . . . are finite and deter- 

 minate, 



,, =/Gr + h) =/(x) +/' (x) h +/" {x) f +/'" ix) ^^ + • • • , 



where /(:r),/' (.r), /" (x), . . . are the vzlues oi /{x -\- h), dujdx, d^u/dx"^, . . . 

 when h = o. This is Taylor's series or theorem. The remainder after the first 

 n terms in /i is expressed by the definite integral 



^ (7« + l/^_l_/^_2)2» 



1. 2.3 ... «J-^ ^ ' ^ 



dz. 

 o 



b. Maclaurin's series. 

 If in Taylor's series we make x = o, and /i = x, the result is 



u =/(x) =/(o) +/' (o) X +/" (o) ^ + /'" (o) -^ + . . . , 



I .2 -^ ^ ' I. 2. 3 ' 



where/(o),/' (o), /" (o), ... are the values oi/(x), du/dx, d'^u/dx'^, . . . when 

 X ^ o. This is Maclaurin's series or theorem. The remainder after the first n 

 terms in x is expressed by the definite integral 



X 



o 



c. Example of Taylor's series. 



«=/(.V + /0 = log(.T + /0. 



/(x) = \ogx, 



du I 



dx X -\- ^ 

 d'hi I 



fiP^) = ^x-\ 



f" (x) = - 



x-% 



dx' ~ (x-\- hy 



Hence for common logarithms, /x being the modulus, 



log {x + h) = log X + /x {x-^ h — \ X-- h- -\- \ x-^ /^8 - . . .), 

 and the sum of the remaining terms is 



h 



1 .2 .^j (x -\- /i — zy 



o 



