[ 265 j 
LIX. Theorems for the Summation of progressional Series. By 
Mr. James BENWELL. 
To Dr. Tilloch. 
Sir, — I HAVE to beg the insertion in your Journal of the fol- 
lowing collection of Theorems, being the algebraic expressible 
sums of the different orders of progressional series respectively 
enumerated and corresponding therewith. ‘The doctrine and 
summation of series is a branch of analytics, it is well known, 
which involves some peculiar difficulties; and the business of in- 
vestigation requires similar and varied artifices of computation 
to be employed, since of the methods and principles devised for 
the purpose, none of them can be so much generalized as to 
embrace a ready and immediate extension on all occasions indis- 
criminately. It is chiefly, or rather solely, by deducing the limits 
and sums of certain orders and forms of series, and by a compa- 
rison of the means used, that we are enabled successfully to ex- 
tend and pursue the same in any new track where the conse- 
quences flowing from their composition and resolution are least 
of all allied and predicable. The theory of series is one likewise 
not very characterized for the precision of its logic; prejudices 
in favour and against the metaphysics of it still exist, and prevail 
with nearly equal force, and these by time alone can be rectified 
and adjusted. bbe: 
Now, as preliminary, making «— 1 =», "| = 2”, n the 
number of terms of the series to be deduced;—then for the 
series 
1420-'+ 52748473 ....(8n—4)x™ to n, terms, 
] —7i —m 
— .(2r+1—3r ”)—(8n—1)e my 
M! , and this 
the general expression is 
by instituting the condition of vanishing quantities in the sense 
: (2x +1) 
convened usually becomes for the limit of infinity -___. 
v 
The sum of the series 
1 4-2a7' + 707* +4 1503... .4(3n?—5n+2)0-™ 
S+v -—m —m v —m 
—.(r—2@ ) + r+ 0%— (3n+1)z, — 5 (Sn? +n)2 
ay 
18 
vt 
Here the limit of infinity will arise when all the terms affected 
by m are feigned practically evanescent, and the same is to be 
implied of all the succeeding cases. 
Vol. 58, No, 282, Oct, 1821. Ll For 
