DIFFERENTIAL AND INTEGRAL CALCULUS. 69 



or 2^ 77-5 > sec' 1 (n + 1 ) - JTT and < sec' 1 (n}. 

 r=-2 r v (r 1) 



So, in general, the sum of the series 



f (l) + f (2) +...+ ' (), 

 lies between the limits 



</> (n) - (0), and (n + 1) - < (1). 



Thus, if 0' (a) = sec' 2 Ox, <f> (x) = 



and sec' 2 + sec' 2 20+... -I- $Qc 2 nd lies between ^ and 



C7 



tan (ft + 1) 9 tan0 ., , ,, * A - Al 



- , provided that sec #0 increase with x 



V 



throughout ; and, therefore, xd < \ir always, therefore 

 (n -{- 1) < ^TT. This method generally gives, Jimits for the 

 sum of any series which are of some value, except when 

 $' (x) is tending to co at one end of the series. Thus, if 

 nQ in the last be near ^TT, the difference of these limits 

 will be large, and they therefore not of much use. 



DIFFERENTIAL CALCULUS. III. 

 1. If y (H- x^ n = sin (n tan' 1 oj), prove that 



and thence that 



2. Prove that if f(x + h) can be expanded in a series 

 of ascending integral powers of h, that expansion will be 



/(*) + hf (x) + f (x) +...+ *!/ (*) +... . - 



Expand sin (x + h) in this manner, and deduce the expan- 

 sions of sin^ and cos^. 



