Mr. stokes, on THE CRITICAL VALUES OF 



course the first n terms of a ilivergent series may be the limits of those of a convergent series ; nor 

 does it appear possible to invent a series so rapidly divergent that it shall not be possible to find a 

 convero-ent series which shall have for the limits of its first n terms the first n terms respectively of 

 the divero-ent series. Of course we may employ a divergent series merely as an abbreviated mode 

 of expressing the limit of the sum of a convergent series. Whenever a divergent series is employed 

 in this way in the present paper, it will be expressly stated that the series is so regarded. 



Convero-ent series may be divided into two classes, according as the series resulting from taking 

 all the terms of the given series positively is convergent or divergent. It will be convenient for 

 the purposes of the present paper to have names for these two classes. I shall accordingly call 

 series belonging to the first class essentially convergent, and series belonging to the second 

 accidentally convergent, while the term convergent, simply, will be used to include both classes. 

 Thus, according to the definitions which will be employed in this paper, the series 



SB + \X^ + 5*' + ••• 



is essentially convergent so long as ,r^ < 1 ; it is divergent when ,v-> 1, and when ai = \ ; and it is 

 accidentally convergent when x= -\. 



The same definitions may be applied to integrals, when one at least of the limits of integration 



/•" sin ,v 

 is OS . Thus, if n > 0, f°.v--dx is essentially convergent at the limit eo , while / — ^ dx is 



only accidentally convergent, and J" sin a; dx, not being convergent, comes under the class of 

 divergent integrals. These definitions may be applied also to integrals taken between finite limits, 

 when the quantity under the integral sign becomes infinite within the limits of integration, or at 



— divergent at the limit 0. 



2. Let /(.») be a function of x which is only considered between the limits ^ = and .v = a, 

 and which can be expanded between those limits in a convergent series of sines of — and Us 



multiples, so that 



fix) = A, sin — + Jo sin ... + A„ sin + (1). 



To determine J.„ multiply both sides of (I) by sin -^ dx, and integrate from a: = to .r = a. 

 Since the series in (l) is convergent, and sin -^ does not become infinite for any real value of x, 



mrx 

 a 

 mrx 

 a 



we may first multiply each term by sin dx and integrate, and then sum, instead of first 



summing and tlien integrating*. But each term of the series in (1) except the w'" will produce 

 in the new series a term equal to zero, and the m'" will produce ^aA„. Hence 



n-TTX 



A„ = - f{x) sm dx, 



and therefore /(.r) = - 2 j f(^v) sin - d.r .sin — 



uttx . . nTTX 



Moigno, Lecons de Caleul Diffirentiel, &C. Tom. ii. p. 70. 



