180 
SERIES OF FUNCTIONS 
f a p n dx = l P|- 
•Jo 2 •Jo dx 
dx 
as n — oo. 
Thus we cannot integrate the F series termwise. 
2. As another example in which the convergence is not uniform 
let us consider 
v J ¿Lf L e (n+X)x e nx J 
0. 
Here 
F = 
nx 
The convergence of F is uniform in 21 =(0, 1) except at x — 0. 
The peaks of the curves F n (x) all have the height e _1 . 
Obviously the area of the 
region under the peaks can be 
made small at pleasure if m is 
taken sufficiently large. Thus 
in this case we can obviously 
integrate termwise, although 
the convergence is not uniform 
in 21. 
We may verify this analytically. For 
CF n dx = C—dx = - - 1 + nX = 0 
^/o */o e nx n ne nx 
3. Finally let us consider 
as n = oo . 
n“x 
Fix') = V I ( n + ^ ~) 2 x 
{ 1 + (n + l) 3 a: 2 1 + vdx 2 
= 0. 
Here 
F n {x~) = 
1 + 'fdx 2 
The convergence is not uniform at x = 0. 
The peaks of F n (x) are at the points x = 7i at which points 
F n = | V n.