Summation of Series. 401 
deduced the fum of any feries, whofe general term is 
az bee ae Bee, 
Bpeetep ic bepaMbeket fat fti.etft2.Rexet gate pike 
[we e* 
And alfo from the fum of the feriefes 1 _—"-» 4 “—- &c. 
eC invetak e+2 
x” I 2 4 
eer ic. 2) ie. ace! be deduced 
7 eee ria &c : aes &c. &c. can be deduce 
the fum of any feries, whofe general term is 
ax bz" 4 &e. 
Qz-e. 2zte+1 .&c.xaztf.oz+f+1.&c.2z+g.9z+g+i. &c. 
The method of adding more terms of a given feries toge- 
ther, as before taught, may be applied to thefe and all other 
feriefes. For example: let the given feries be 14+3¥+4%" 
+ 4x° + &c.; add twotermsconftantly together, and it becomes 
Riess 

2 A 2A Bee 
peer &c a a, et &ces celia Guna es) +. 
2 ey 5° 2, Bid 
GSA -  az+2+(22+4 1) 
= +&c. whence the general term is se 
rare From the methods before given of addition, fub- 
traction, and multiplication ; and the feriefes found by this 
method, can be derived feriefes, whofe fums are known. 
12. Suppofe a given feries ax" -+ burt: + curt + devts + &c. 
whofe fum # is either an algebraical, exponential, or fluential 
fluxion of x; multiply the equation pa axt + bart + cx + 
darts 4 8c. into wt *,andthererefults avt—"p = ant” 4 bab + 
ext 4 &c.; find the fluxion of this equation, and there 
follows — multiplied into the fluxion of the quantity (xt—"9) 
a 
= traxtar4 (sorchs) bx 4 (sere 2s) cbt + &e. 
of which the general term is (=r=:%s) x ft, where = denotes 
the diftance from the firft term of the feries, and ¢ 
You. LXXIV. Gegeg eas 
