Rev. G. Phillips ow the Summation of Series. *4-l 



The general expression for the first n-l' series of the se- 

 cond member of this equation is 



1 ^ +&C. 



Hence if we can sum the series for k squares, we can for 

 yt+l; we have summed it when A: = 1 &c. .-. we cantor 

 A- + 1 or 2, &c. .-. for 3, and generally. 



If we pursue the methods here adopted, we may sum me 

 series 1 1 + &c. 



1"'.2". 3^... yfc* 2™. 3".4P...X+P' 



where ?m, n,p, &c. may be any numbers whatever; the process, 

 however, will be much too tedious to attempt the operation. 



Again, X^(^) = K^ + ^ ^ 4 + ^^O 



Jy yJy yJy -y ^y/n^ -y) 



~" l^ 1*" 3"' 5*" / 



&c. &c. 

 We see clearly that from this expression if we go through 

 the preceding articles with reference to the odd numbers, we 

 shall have the sums of the several series : 



_i_ + __^ + —1- + &c. 

 !»». 3 3'".5 S*". 7 



_i_ + _L_ + 1 + &c. 



1.3" 3.5"" 5. 7*" 



_JL_ + -1- + -i- + &c. 



l". 3« 3"*. 5" 5"*. 7" 



^ ^ -I- — _ + &c, « an odd number 



1 "*. 11 3"*. « + 2^ S"*. « + "t^ 



1 __ 1 I 



I". S.sTnn+T''*' 3". 5. 7... « + 3' 5™. 7.9... n + 5* 

 + &c. n an even number. 



1 , __JL___ . _i-.==^ + &c. 



^■^ 3.5.7..."^T3''^5.7.9...n + 5* 



. + ^-=^ + he. 



l«.3«..5*...n+l'* 3».5*.7^..n+^* ^ 5».7^9«...n + 5»« 



2Sr. S. Vol. 1 1 . No. 66. June 1832. 3 L LXI. No- 



