ii'lO THIRD REPORT — 1833. 



approximates continually to — 1 as a limit, and the terms be- 

 come all positive or all negative, according as tlie first negative 



coefficient is that of an odd or of an even power of — . It follows, 



a 



therefore, that if a he greater than b, the series will be conver- 

 gent and finite in all cases ; if a be equal to b, it will be 0, 1 , 

 or 00 , according as ?i is positive, 0, or negative ; and if a be 

 less than b, it will be infinite. 



The occurrence of the last of these signs or values is an in- 

 dication generalhj that some change has taken place in the na- 

 ture of the quantity expressed by (« — by, in the transition 

 from a'!> bto aK b, which is of such a kind that the correspond- 

 ing series is not competent to express it : thus, if n = ^, then 

 (« — b)" is affected with the sign \/ — l when a is less than b, 

 whilst no such sign is introduced nor introducible into the equi- 

 valent series corresponding to such relative values of a and b : 

 and a similar change will take place, whenever a transition 

 through zero or infinity/ takes place. 



In this last case (a — 6)" would appear to attain to zero or in- 

 finiti/, but not to pass through it, and no change would appa- 

 rently take place in its affection corresponding to the change of 

 affection of a — b; but the corresponding series will under the 

 same circumstances change from being finite to infinite, a cir- 

 cumstance which we shall afterwards have occasion to notice, 

 and which we shall endeavour to explain in the course of our 

 observations upon the subject of diverging and converging 

 series. 



In the preceding examples the sign or symbol oo has not 

 presented itself immediately, but has replaced an infinite series 

 of terms, whose sum exceeded any finite magnitude ; and it may 

 be considered as indicating the incompetence of such a series 

 to express the altered state or conditions of the quantity or 

 fraction to which it was required to be altogether, as well as 

 algebraically, equivalent. In the examples which follow, it will 

 present itself immediately and will be found to be the indica- 

 tion of a change in the algebraical form of the term or terms in 

 which it appears, or rather that no terms of the form assigned 

 can present themselves in the required equivalent series or 

 expression. 



The integral I x" d x = ^ + C is said to fail when 



'^ J 71+1 '' 



n =■ — 1, in as much as it appears that under such circum- 



/r" ~ ' 



stances r becomes co , which is an indication that the va- 



« + 1 



