Convergence/ of Fourier 's Series, 135 



into (m + 1) elements as above, without transforming the 

 variable we can show as before that each element vanishes 

 when ?i = oo . But the sum of the m elements taken in this 

 form is not determinate when n = x> . For as n increases 

 without limit, m also increases without limit, and therefore 

 the sum tends to the indeterminate value go x 0, as in the 

 case of any definite integral. We have thus no means of 

 determining whether A and C vanish when n = co . But by 

 means of the transformation (2n+ 1)^0 = $, we see that each 

 element is really of the form 



^Jfx(Wri) sin< ^- 



ir(2n+l) 

 Here the integral, independently of the facto , is 



infinitely small when n = x> , and this multiplied by 



7r(2rc + l) 



gives us an infinitesimal of the second order. Hence the 

 sum of the m elements is not really (go xO), but (go x O 2 ), 



or ( — X V and the form — when looked into is found to be 



v 30 ' m °° 



derived from ^ ^ , whose real limit is < 1 . It is this that 



2n + V 



determines the convergence of S w to its limiting value. 



23. It is necessary to remark that in general an element of 



4:7T 



the integral S n in which the range of integration is 9 , v 



vanishes when n = co only when 6 is, numerically, not less than 

 h } and h is not less than the value necessary to ensure that 



p and I are Dotn infinitely small, t being 



= or < ^ 7 (see 15). Of course, since t can be diminished 



2n +1 v ' 



without limit by increasing n without limit, and F(0 + ^) 



is continuous, this condition can be satisfied for values of h 



less than any assignable finite limit, however small. But as 



n increases without limit, the two infinitesimals t and h must 



diminish at different rates ; for whereas t tends to the value 



zero at a constant rate, h must do so at a constantly dimi- 



nishing rate. Thus, t being « — — i? h may be ; , &c. The 



consequence of this is that in the integral 



I 



Vtf'+yaM&Mi*, 



9 



