EXPOSITION OE THEORY OF POWER SERIES. 
219 
But it is easily seen that the converse proposition is not true — 
that is to say, it is not sufficient that v have a limit in order that S 
may be convergent, as the well-known example — 
log. 2 = 1 — i + %- 
+ 
+ 
sufficiently proves. 
2 n- 
-1 2n + 
In the first case — viz., when S is convergent — the series v is said 
to be absolutely convergent . 
Any convergent series whose terms are all positive integers is 
absolutely convergent. 
According to Biemann (Ueber die Darstellbarkeit einer Function 
durch eine trigonometrische Heihe, § 3) Dirichlet was the first to clearly 
perceive the great difference which exists between series absolutely 
convergent and series convergent only; but to Eiemann himself 
(Joe. cit., § 3) belongs the honour of having put that difference in a 
very forcible light. 
In the former series (absolutely convergent) the order of the 
terms may be altered in any manner whatever without changing the 
value of that series, but the case is quite different with regard to series 
convergent only ; for as Biemann has pointed out — we 7 iiay 7 naJce any 
series convergent only , take any value roe please , by arranging the terms 
i7i an appropriate order. 
This statement is based upon the simple fact that (using the 
notations above) — 
lim. P M = go 
and also lim. Q w = go for n = go 
Then let — 
* a t a 2 a 3 .... a p 
and b 1 b 2 b 3 b q . . . 
be respectively the positive and the negative terms of aiiy of those 
series arranged in decreasing order, and let C be any quantity what- 
ever (supposed positive to fix the ideas). Take just enough positive 
terms to bring their sum above C, then add just enough negative 
terms to bring the sum below C ; again add just enough positive 
terms to bring the sum above C, etc. We can repeat this operation 
indefinitely, since — 
lim. P n = go 
lim. Q rt = co 
and therefore form a series. But the difference (in absolute value) 
between one of the above expressions and C never exceeds the term 
(in absolute value) preceding the last change of sign, and, as owing 
to the convergence of the series under consideration, 
lim. a p — 0 lim. b q — 0 
P = CO (jj — go 
