936] 
NOTE ON UNIFORM CONVERGENCE. 
343 
the series is 0 + 0 + 0 + ... We have here an instance, and there is in fact a dis 
continuity in the sum, viz. x < 1 the sum is 
(1-«)(1 + x + x % + ...), =(1-#).—— , =1; 
x CO 
whereas for the limiting value 1, the sum is 0 + 0 + 0 + ..., =0. The series is thus 
uniformly convergent up to and exclusive of the value x—\, but for this value there 
is a breach of uniform convergence. 
I remark that Du Bois-Reymond in his paper, “Notiz liber einen Cauchy’schen 
Satz, die Stetigkeit von Surnmen unendlicher Reihen betreffend,” Math. Ann., t. iv. 
(1871), pp. 135—137, shows that, when certain conditions are satisfied, the sum cf>x is 
a continuous function of x, but he does not use the term “ uniform convergence,” nor 
give any actual definition thereof. 
M. Jordan, in his “ Cours d’Analyse de 1’Ecole Polytechnique,” t. I. (Paris, 1882), 
considers p. 116 the series s = u 1 + u 2 + u 3 +..., the terms of which are functions of a 
variable z, and after remarking that such a series is convergent for the values of z 
included within a certain interval, if for each of these values and for every value of 
the infinitely small quantity e we can assign a value of n such that for every value of p, 
Mod (u n+1 + u n+2 + ... + u n+P ) < Mod e, 
e being as small as we please, proceeds:— 
“Le nombre des termes qu’il est nécessaire de prendre dans la série pour arriver 
à ce résultat sera en général une fonction de z et de e. Néanmoins on pourra très 
habituellement déterminer un nombre n fonction de e seulement telle que la condition 
soit satisfaite pour toute valeur de 2 comprise dans l’intervalle considéré. On dira 
dans ce cas que la série s est uniformément convergente dans cet intervalle.” 
And similarly, Professor Chrystal in his Algebra, Part II. (Edinburgh, 1889), after 
considering, p. 130, the series 
+ 
x +1.2x + 1 2x +1.3x + 1 
+ ...+ 
(il — 1) X + 1 . nx + 1 
+... 
for which the critical value is x = 0, and in which when x — 0 the residue R n of the 
series or sum of the (n + l)th and following terms is =——- proceeds as follows:— 
1X00 “f“ JL 
Now when x has any given value, we can by making n large enough make — 
smaller than any given positive quantity a. But on the other hand, the smaller x is 
the larger must we take n in order that —may fall under a; and in general 
ixoo "f* JL 
when x is variable there is no finite upper limit for n independent of x, say v, such 
that if n > v then R n < a. When the residue has this peculiarity the series is said to 
be non-uniformly convergent; and if for a particular value of x, such as x — 0 in the