220 
PROCEEDINGS OF SECTION A. 
that difference must be infinitely small — that is to say, C will be the 
limit of the series we have formed. A very simple example will make 
this clear. Take 
log. 2=1-*+*-*+ 
+ 
= n r(-±- — — + — 1 -M 
n-t\4in — 3 An — 2 An - 1 4 n ) 
and also 
+ + + ... + 
4m — 3 4^ — 1 
1 
2n 
log. 2 and X are series differing only in the order of their terms. 
Make the difference — 
x ~ lo s- 2 = M (aS=I-5j 
or X — log. 2 = * log. 2 so that 
X = f log. 2. 
Having now emphasised the importance of the separation of series 
into absolutely convergent and convergent, we come to the main part 
of this paper. 
FUNCTIONS REPRESENTED BY MEANS OF SERIES. 
And first of all : Under what conditions may a series represent a 
continuous function ? 
Cauchy, in his “ Cours d’ Analyse Algebrique ou de l’Ecole 
Poly technique,” had enunciated the following proposition : — - 
“ When the different terms of the series 
n t + u 2 + ... * + u n 4 - u n + j + .... 
are functions of a single variable x, continuous with regard to this 
variable, in the neighbourhood of a certain value for which the series 
is convergent, the sum of the series is also, in the neighbourhood of 
that value, a continuous function of x.” 
Abel, in a footnote of his well-known “ Memoir© sur la serie — 
_ m m (m — 1) , , 
1 + y 9 + 2^2 — X + 
[written in French by Abel during the winter 1825-26, translated into 
German by Crelle and published in his journal in the month of 
February or March, 1827 — See “ CEuvres Completes d’Abel,” edit. 
