1821.] a Method of applying Maclaurin's Theorem. 105 



The student must understand that the developement of a func- 

 tion by the preceeding method is not always the most conve- 

 nient ; for instance, Examples 12 and 13 may be answered in a 

 more simple and general manner by note N (with some slight 

 alteration in the symbols), taken from the translation of Lacroix' 

 Differential and Integral Calculus. 



X3 x^ Xl X9 



^ , . ^ Sin. o: 1.2.3 1...5 1...7 1...9 



Per trig. tan. x = 



COS. ar , x* x* x* j* 



+ •; r — &C. 



1 . 2 1...4 1...6 1...8 



radius unity. 



By writing « = 7^, ^ = j-jt^t^, c = ~^, d = -j-l^, e = 



r — r, f = - — r, &c. the above equation becomes tan. x = 



a: — ft x3 + d x^ —fxi + h a;» — &c, 

 1 — a x^ + c X* — e x^ + g X* — &c. ' 



Now from a little attention to these equations, it will appear 

 that tan. x may be represented by the series A x + IB x^ + Cx* 

 + D^7 + E:i9 + &c. .......... 



that is, fL^llLLl^LlLf^^ ^s 



' I — a x^ + c X* — e x^ + g x** — &c. 



4- D X? + E .r9 + &c. 

 From whence we get '^ •^'*'- ^'^ 



(B-Aa)a;' 

 a:— ^>j:* + <?a;*— /a:7+ Aa;^— Scc.= ^ 



(C-Ba + Ac)a:5 ^ '^ ^i 

 (D-C« + Bc~A^^y:i^^ 

 (E-D« + Cc-Be + Ag)x» 



&c. 



Then by equating the coefficients of the corresponding powers 

 of a? on each side of the last equation, we have 

 A = 1. 



2 



B = Aa-C = 



1.2. 3' 



16 



= B«-Ac+c^ = 



...5' 



B =:Ca-Bc + Ae - f = ~, 



E = Da-Cc + Be-Ag + ^ = j^, - -^ '^ 



8cc. ..; 



The law of continuation being evident, we, therefore, have 



, . 2x3 . 16 x5 272x7 . 7936 x» . , q 



*^";*' = ^ + no + r75 + T31 + tttt + ^«- 



The same method may be applied to the developement of the 



