326 REPORT — 1861. 



n=Qo , 



Observing that lim p S — — — is intermediate between 



,. T" dx J ,. r*" dx 

 lim p I — — and lim p I , 



and is consequently unity, we infer from the last inequalities that 



lim a 2 , — — , 



«=N+lW 



and therefore also »=oo , 



lim p S , , 



n=L *»^+'' 



which is identical with it, because 



limpS — !_=0 



n=l «'+" 



«=oo , 



differs from a bv a quantity comminuent with S; i, e. limpS ■ =g, 



n=l «» '^'' 



since by hypothesis ^ is a quantity as small as we please. 



(ii.) A convergent infinite series may be convergent in two very different 

 ways. It may be convergent, and always have the same sum irrespective of the 

 arrangement of its terms ; or it may be convergent for certain arrangements 

 of its terms, giving the same or different sums for these different arrangements, 

 and divergent for other arrangements. We suppose, however, that we con- 

 sider only such different arrangements of the terms of a series as are compa- 

 tible witii the condition that any term which occupies &Jinitesimal place in 

 any one arrangement should occupy aJiTiitesimal '^\sice in every other arrange- 

 ment*. Thus the series 



1 1 1 



JI+P+2H-P + 31+P+ •••> p>o, 



is convergent, and has the same sum in whatever order we sum its terms ; 

 but of the two series 



2* 3' 4' 5^ 6^ 



1+11+1+11 + ... 



3* 2^ 5* 7^ 4* 



* This condition is necessary, because without it the sum of no series whatever would be 

 independent of the arrangement of its terms, if by the sum of a series we understand the 

 limit to which we approximate by the continual addition of its terms in the order in which 

 they are given. For example, the series cited in the text, 

 _j_ _1_ _i_ 

 ji+p+.^i+P+3i+P+ • • • . P > 0, 



is convergent, and its sum is irrespective of the arrangement of its terms, provided that 

 arrangement satisfy the condition enunciated in the text. But if we were to arrange the 

 terms of the series in an order regulated (say) by the number of primes dividing their deno- 

 minators, the limit to which we should continually approach by adding together the terms 



1 1 



taken in their new order, would be 2 "frr, in which p denotes any prime, and not S ,_).p, 



in which n denotes any integer. 



