CEUCES MATHEMATICS. 



21 



which expresses B„ in terms of the numbers of lower orders. 

 Griving consecutive values to n, 



(_)7.+ ]j5„ 



2n — 3 - 2n — 1 



2n — 5 2n — 3 



2?i— 1 

 2(2n + 1) 



2« — 3 

 2(2n— 1) 



- A + ÏQB, 



Whence by determinant elimination and a little reduction we obtain 



B.. = h 



n— 1 



2n — 3 2n- 



2m- 



B, 



1 

 ■J 

 3 



s 



T 



2n- 



2n— 1 

 2w — 1 



2k + 1 



Expansion of the Invebse Tangent without the Use of Limits. 



Expansion of tan ~^ x without the use of limits. 



As in the expansion of sin ^, a little consideration shows that the expansion of 

 tau~' X must have its first term x and must contain only odd powers of x. 

 Therefore assume 



Then 



and 

 But 



inn-' x — x-\-a,x' + a,,ê-\- a2„+ix-2''+i + 



iaxr' y = y + a,y^ + aAf + a2a+\y-"+^ -\- . 



tan"^ j; — tan"' y = a; — y-'!-a^{x'' — /') + f^-jn+iC^--"'''^ — 2/^"+') 



tan 'x — tan ' y = tan"' -^ — ^ 

 •^ 1 + Xh 



+ xy' 



X 



— ^-j/-i-flp— j/V «2"+!^^— y^"''"^^ 



