EXPOSITION OF THEOEY OF POWEE SEEIES. 
221 
1881, p. 220] points out that Cauchy’s theorem is obviously subject to 
some exceptions, as the series- 
sin. x — i sin. 2x -f | sin. 3x — . . . . 
representing 
— TT < X < 7T, 
is no longer equal to that function 
when x tends towards 7r, since the limit of the series is zero, whilst 
X = n 
(See also a letter from Abel to Holmboe, 16th January, 1826 ; (Euvres 
completes, p. 256.) 
Cauchy, who was so much engrossed with his original work that 
he had seldom time to read any outside publication, did not answer 
Abel’s objections till 1853. But in the “ Comptes rendus de 
1’ Academic des Sciences, 14 Mars, 1853,” in a “ Note sur les series 
convergentes dontles divers termes sont des fonctions continues d’une 
variable reelle ou imaginaire, entre des limites donnees.” He corrects 
very fully his error, and replaces his theorem by the following: — 
“ If the several terms of the series — 
(1) -f* u-2 + .... + + u n + 1 + • • • • 
are functions of the real variable x, continuous with regard to that 
variable between certain given limits, and if also the sum 
r H ' = u n+ i -f . . . + u n > 
becomes always infinitely small for very great values of the positive 
integer n, and infinitely great values of n f > n, the series (1) will be 
convergent, and between the given limits a continuous function of the 
variable x. 
The demonstration of the theorem is immediate, and is based 
upon the fact that the series is equal to the sum of its n first terms 
S n (n being as great as we please, but finite), and the limit of the rest 
r n > for n! = oc. By reason of the conditions given, S» is obviously 
a continuous function of x, whilst nothing can be said a priori 
about r n >. 
But, prior to Cauchy, the question had been thoroughly sifted by 
the German mathematician, Seidel. [Note. — “ liber eine Eigenschaft 
von Beihen welche discon tinuirlicho Eunctionen darstellen ’ ’ — 
Abhandlungen der Bayerischen Akademie, 1847-49.] Not only does 
Seidel give the result afterwards rediscovered by Cauchy, but he goes 
more deeply into the nature of the series considered, and demonstrates 
the following proposition : — 
“ If the terms of a convergent series are continuous functions of 
the variable x y but the series itself represents a discontinuous function 
of x , there will exist in the immediate neighbourhood of the point 
where a break occurs in the value of the function (z.e., where the 
function is discontinuous) some values of x which will make the series 
converge as slowly as we please.” 
