EXPOSITION OF THEORY OF POWER SERIES. 
223 
pupils, Heine, in a memoir, 44 Ueber trigonometrische Beihen” 
[February, 1870 — published in Crelle’s journal, Band 71, p. 353], 
says : — 
“ Until recently it was believed that the integral of a convergent 
series, whose terms were finite between the finite limits of integration, 
was equal to the sum of the integrals of each term, and Herr 
"Weierstrass was the first to point out that, in order to demonstrate 
this theorem, it is required not only that the series converges between 
the limits of integration but still converges uniformly (in gleichem 
Grade con vergire) . ” 
At the same time, however, Heine remarks that, in a paper 
previously published in the volume GO (18GG a.d.) of the same journal 
(Crelle’s journal), Thome # had regarded the notion of uniform con- 
vergency as already acquired. It was seen later on that the priority 
belonged to Seidel’s work. In relation to his theorem it must not be 
thought that infinitely slow convergency always involves discontinuity. 
P. du Bois-Keymond, in “ Beweis dass die Coefiicienten,” etc. — 
Abhandlungen der Bayerisehe Akademie, Band XII. (footnote, p. 119) 
— has shown by an example that a series may represent a continuous 
function of x during an interval when the convergency is infinitely 
slow; a fact shown by Seidel (Joe: cii\, p. 393). This example is also 
reproduced in Chrystal’s Algebra (Part II. , p. 131). To complete 
the history of these investigations we will add that Duhamel had also, 
after Cauchy, considered the same question (Comptes rendus, etc., 
1853, p. 643). 
It results from the above considerations that it is not admissible 
to consider a convergent series as representing a continuous function 
of x — even when its terms are continuous functions of x — without 
careful examination. 
The following theorem embodies all our knowledge on the subject 
of the representation of continuous functions by series : — 
THEOREM I. 
Any function/* 0) represented by the series — 
/ O) = O) + (x) + . . . . + u n (x) + .... 
(convergent for a < x <b) where the terms are continuous functions 
of x w r ill be itself a continuous function of x in the neighbourhood 
of any value x 0 of x (between a and V) if the series is uniformly 
convergent in the neighbourhood of that value x 0 . 
c being an infinitely small quantity, we have — 
/ O o + e ) — / Oo) = u t (x o + € ) + . . . . + u n (x o + e ) — 
K Oo) + . ‘ • + «» 0*0] + (a?o + O — Oo)- 
As each term of the series is a continuous function of x in the 
neighbourhood of x 0 — 
U * 0*0 + 0 + . . . . -h Un (#o + 0 — [u t (x 0 ) +.... + tin Oo)] 
is infinitely small (n being finite) ; and if / O c ) is assumed to be 
uniformly convergent in the neighbourhood of x Q , B n (x Q + e) and 
0 o ) may be taken — (each separately) — infinitely small, wherefore 
f Oo) will be then continuous in the neighbourhood of x = x Q . 
* “Ueber h y peygeometrisch e Kettenbriiche,” p. 334. 
