EXAMPLES OF THE EXPANSION OF FUNCTIONS. 97 



12. 



13. 

 14. 



16. For what values of x does Taylor's Theorem fail, if 



*/((x-a) 7 (x-br\ ,,.,..,, ,.. 

 w = . / 4- - / xa -- ri an d which is the first differential 



VI (^- c ) J 



coefficient that becomes infinite ? 



17. Shew that 



W 

 tan' 1 (a? + A) = tan l x + h sin 2 ^ - sm s & sin 2^ 



h 3 h* 



+ sin 3 ^ sin 30 - sin 4 ^ sin 40 + ... 

 o 4 



TT 



where 6 = -- tan" 1 ^. See Example 18 of Chapter V. 



21 



18. By putting h = x in Example 17, shew that 



TT ., cos*0 sin 20 cos 3 sin 3^ 



cos# H -- - -- 1 



O Q 



cos 4 sin 40 

 4 



19. By putting h = x in Example 17, shew that 



7r_sin0 sin 20 sin 30 sin 40 

 2 cos 2 cos*0 3 cos 8 04 cos 4 



20. By putting A = V(l +%*) in Example 17, shew that 



1 . 1 . 1 . 1 



8 '* ' 2 S1 3 S1 4 S * 



T. D. C. U 



