VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 315 



Cor. 2. Hence also -f- = - — - + -— - + &c. ad infinitum. 

 2o v v v v 



and i == _;_i_!_"'--L.- — &c. ad infinitum. 



2t> o o 



p RO p i — If — * — be the general term of a series formed by writing for n 



rn + m ° 



any series of numbers in arithmetic progression, and whose signs are alternately 

 -f and — ; then if a series be formed by collecting two terms into one, begin- 

 ning at the first term, the sum of the series thence arising will be less than the 

 sum of the given series by — . If a series be formed by beginning at the second 



term, the sum of this will be greater than the sum of the given series by — . 



■p or i e t — " " "I" a — be any two successive terms of the series, which, 



r Ul 1CL r „ + m {n + a) r + m J ' 



if we begin to collect at the 1st term, that term being +, will be two terms to be 

 collected into one, and which will therefore give (r „ + m) x ~[™ + a) ,. + „,-, for a 



general term of the resulting series. Let us now make n infinite, and then the 

 denominator of this term becomes infinite, and the numerator finite; therefore 

 the terms of this latter series at an infinite distance becoming infinitely small, the 

 series will there terminate. Now, by making n infinite in the given series, the 



two successive general terms at an infinite distance become ; ; consequently 



this series is still contiuued after the other terminates; and the terms of such a 

 continuation will be (as they begin with 



__2 n + a — , making n infinite) 1- -j- &c. which will 



rn + m (n + a) r + m J b ' r r r r 



also be continued ad infin. and whose sum by the lemma is — ; consequently the 

 given series exceeds that which is formed by collecting two terms into one, be- 

 ginning at the first-, by — ; hence the sum of the latter series + — will be equal 



to the sum of the former. If we begin to collect at the 2d term, then will — 

 ii + a 



~ 1 _ - be the two successive general terms of the given series to 



m + m ' (n + a) r + m ° . 



be collected into one; consequently the continuation of the given series, when 



n becomes infinite, will be 1 + - — &c. ad infinitum, whose sum 



by cor. 1 to the lem. is ; in this case therefore, the sum of the given series 



is less than the sum of the series formed by collecting 2 terms into one, begin- 

 ning at the 2d term, by ^-; hence the sum of the latter series — — will be equal 

 to the sum of the former. 



1 2 3 4* 



Exam. Let the given series be - — - + - — - + &c. 



Here r = 1, n = 1, 2, 3, 4, &c. and m = 1. Now, if we begin to collect 



s s 2 



