218 
PROCEEDINGS OF SECTION A. 
The terms of the series may be positive, negative, or more 
generally complex quantities* 
In all cases of use of series, the first matter to be ascertained is 
whether the infinite succession of terms represents a definite quantity 
or not. 
Considering first a series — 
u u x +■ + .... 4- u n 4- .... 
having for its terms positive quantities only — viz., in fact, numbers — 
the number — 
2 u n = Ul + u 2 + .... 4- u n . 
increases with n; and two cases are to be considered. Either 2 u n has 
a limit, in which case the series represents precisely this limit; or it 
has no limit, and in that case the series represents nothing definite. 
As is well known, the series in the first case is said to be 
convergent , in the second case divergent. 
And these definitions are naturally extended to any kind of series, 
that is to say : A series is said to be convergent when the sum of its n 
first terms has a limit for n = oo , divergent in the contrary case.f 
Convergent series having positive and negative terms — 
v = v t + v, + .... + v n + .... 
which we come now to consider, are to be divided into two great classes, 
according as the series formed with the terms taken in absolute value 
is convergent or divergent. 
Denoting by Y r the absolute value of v r — that is to say, the value 
of the term v r independently of its sign (modulus, tensor) — if 
s = v, + v 2 +....+ v„ + ... . 
is convergent, it can be proved as follows that v is convergent^ : — 
Denote by P rt the sum of all the positive terms to be found 
amongst the n first terms of the series v and by Q re the slim of the 
negative terms, taken in absolute value, of the same n first terms of 
the series. 
X = + .... + Vn = P n — Qn 
and also S w = V x + V 2 4- . . . . 4- Y n = P n 4- Q» 
As S 7i has by hypothesis a finite limit, neither P n nor Q n can be 
infinite at the limit (it = oo ) ; or in other terms P H and Q„ must have 
separately a limit for n = co . 
Let — - lim. P w = P 
lim. Q„ = Q, 
then — 
lim. 2„ . = P — Q 
“ Q.E.D. 
* In the present paper I will not consider the latter series. 
f According to these definitions, what is called here convergent series includes the 
convergent and semi-convergent series of certain authors, and what is called divergent 
series includes the divergent and oscillating series. 
X Although this demonstration is well known, I have thought it necessary to give 
it again, in order to emphasise certain points which are not perhaps sufficiently 
insisted upon in ordinary treatises. 
