DIFFERENTIAL AND INTEGRAL CALCULUS. 65 



These series may be written 



(sin" 1 x} 3 _ 1 x 3 1.3 /!_ l\ of 

 [3 ~ 2 3" + 2l " (^ + 3V 5" 



, L^fl.I L-l^^i 

 "*" 2.4.6 U 2 3 2 T 5V 7 



(sin" 1 a-) 4 _ 2 J_ of 2.4 /_! _1\ a 8 

 [4 ~ 3 ' 2 2 ' 4 + 3^ ' 1 2 s + 4V 6 



2.4.6 /^ _! 1\ ^ 



The series for sin#, cosa?, tan l x are found in all works on 

 Plane Trigonometry, and can of course readily be ob- 

 tained by Maclaurin's series ; tan' 1 x can however be easily 

 found by integration, thus 



-j- (tan' 1 x) = - = 1 a; 11 + x* x* +...to oo (when x < 1) ; 



dx^ 1 + x 



~3 ^,5 ^,1 



therefore tan' 1 x = 



and since when o? = 0, tan" 1 a? = 0, (7=0; or the series is 

 completely determined. Of course tan" 1 (x) must be an 

 angle between %TT and |TT, and since, for the series to be 

 convergent, x< 1, tan" 1 ^) must lie between - ^TT and ^TT. 

 The following functions of x will furnish good examples 

 for the student : 



(1) i-LM 



sin (n tan" 1 x) 



giving in (3) the remainder after the first n significant 

 terms. 



Also tan' 1 (x + h) can be expanded in terms of 7z, since 



( -7- ) (tan" 1 x} has been found. 

 \dxj 



The geometrical proof of the equation 



</> (a; -1- 7i) = < (a?) 4 /?(/>' ( -f ^A), 



