92 EXPANSION OF FUNCTIONS. 



Hence we have 



35 n 



f 1 \ n y n+1 ( ( n -f. 1 ) TJ- 



_8i n jv_ ( n + 1 ) 



(n + !)(! + 0V) 



And if a; be numerically less than 1, the last term can be 

 made as small as we please by sufficiently increasing n \ so 

 that the infinite series 





can- by taking a sufficient number of terms be brought as near 

 as we please to tan -I ic. 



120. Expand sin" 1 ^ in powers of x. 

 Assume suT l x = A + A 1 x + A 2 x y + ... +A n x n + ....... (1). 



Differentiate both sides ; thus 



^ l _ "^ = A 1 + 2A,x + SA s x* +... + nA n x n ~ l + . . . (2). 



*n i 



But 



by the Binomial Theorem. 



Hence, comparing the coefficients in (2) and (3), we de- 

 termine A I} A z , ..., and putting x = Q in (1) we get A =0. 

 Substituting in (1), we have 



la; 3 1.3 

 sm ^= 



It should be remarked that there are two considerations 

 which limit the generality of this investigation. We take 



T- - rr as the differential coefficient of sin" 1 x. whereas the 

 V (l-) 



radical ought strictly to have the double sign : see Art. 65. 

 And we take sin" 1 x to vanish with x, whereas we know, by 

 Trigonometry, that sin" 1 x might be any multiple of TT when 

 x vanishes. 



