REPORT ON CERTAIN BRANCHES OF ANALYSIS. 279 



irregular, and after the third term they increase with great 

 rapidity. The series under consideration, therefore, even for 

 considerable values of s, becomes divergent after a certain 

 number of terms. But an approximate value of the series will 

 be obtained from the aggregation of the convergent terms only : 

 and it has been proved by a German analyst* that the error 

 which is thus made in the value of the generating function will 

 in this case be less than the last of the convergent, or the first 

 of the divergent, terms. 



It has been usual amongst some later mathematicians of the 

 highest rank to denominate diverging series, without any di- 

 stinction of their class, as false, not merely when arithmetical 

 values are considered, but also when employed as equivalent 

 forms, in purely symbolical processes. The view of their ori- 

 gin and nature which we have taken above would explain the 

 sense in which they might be so considered in relation both to 

 arithmetical processes and to the calculation of arithmetical 

 values. It seems, however, an abuse of terms to apply the term 

 false to any results which necessarily follow from the lavys of 

 algebra. M. Poisson, perhaps the most illustrious of living 

 analysts, has referred, in confirmation of this opinion, to some 

 examples of erroneous conclusions produced through the me- 

 dium of divergent series f ; and as the question is one of great 

 importance and of great difficulty, I shall venture to notice them 

 in detail. 



Let it be required to express the value of 



»+i d X 



=/: 



{{\-2ax + a^){\ -2bx + b^)Y 



{ 



by means of series. 



Assuming K = (1 - 2 « x + «2)- 4 and K' = (1 - 2 5 r + b^ *, 

 let us suppose K and K' developed according to ascending and 

 descending powers of a and b respectively ; or, 



K = 1 + a X, + «'■' Xg f a^ X3 + &c. 



K' = 1 + 6 Xj + 62 Xg + b^ X3 + &c. 



K' = l + ^Xi + lx,+ lx3+&c. 



* Erchinger in Schrader's Commentatio de Siimmatione Seriei, Src. Weimar 

 1818. 

 t Journal de I'Ecole Polylechnique, torn. xii. 



