224 
PBOCEEBIims OP SECTION A. 
THEOREM II. 
The series formed by the integrals of the terms of f(x) will 
represent the integral of f(x) during any interval where f(x) is 
uniformly convergent ; the limits of integration being included between 
those of uniform convergency. 
We may write — 
f{x) = + U % {x) + .... + Un (#) + Rn (#)> 
therefore, as the number of terms is limited — 
Pfi Cft , Pi 
/ f (x) dx — / u t (#) dx + / 
J a J a J a 
u 2 (x) dx + . . . 
( x ) dx + / R n (#) dx 
where a and /3 are included between the limits a and b of uniform 
convergency — 
a < a < /J < b. 
Now, from the theorem of mean values- 
r 
R* (x) dx = (/3 — a) E n (£) 
where £ is a value of x between a and — and as R„(f) may be taken 
infinitely small from the hypothesis — 
E- n (x) dx may 
be also taken 
infinitely small ; 
that is to say, the series- 
*J8 
u x {x) dx 
la 
/: 
u 2 (x) dx + 
is convergent,^ and represents- 
r 
f(x) dx 
We come now to the special case of series arranged in order of 
increasing powers (whole and positive) of the variable x, such as series 
obtained by means of the so-called Maclaurin's formula. 
Let— 
u = a Q + i x 4- tf' 3 x 7 + • * • • + ct n + 
be such a series. 
* The necessary and sufficient condition forthe convergence of a series is that the 
sum of any number of its terms after the wth be infinitely small. 
