Summation of Series. 493 

of 
into the terms of the feries 1, ee ie paPel, &c. and a feries is 
‘ aie 
um m ny 
Dene acme ae prptixm.m—n Nig ear 
deduced a +-——a Cane Sata eet + &c. 
ut 
e ° ° ——+— . 
whofe fum is known, if the fum of the feries=a+% 7 is 
known. 
Ex. 2. If the feries begins from the /41" term of the above- 
s é : MB Ap ie 3 
mentioned binomial theorem a — as ae "x+&c. viz. the 

m—1+1 m—l+2 x] 3 
re be es a ee 
l+2n @ l+3n a l+4n a3 
ef which let the refpeftive terms be multiplied into 1, 
£ s at » &c. there will refult a feries whofe fum is known. 
Tease 
Ex. 3. From the rule firft given by me for finding the fum of 
the terms at 4 diftances from each other, the fum of the feries 

pa Mate y mal an i m—ie hn x + Pox mak b+tn 
l+2.H# l+3n at GP l+h+an 
ihe We 2 rE 2 m--lf2hns Be, where P denotes the co-effie 
t+h+3n /+2h+ in a 
cient of the preceding term, can be deduced; and confequently 
‘the fum of the feries deduced from multiplying the fucceffive 
A G A aire Dp I 
terms of this feries into the quantities 1, t,t ue pei, &c. res 
fpectively. 
The general principles of this cafe wete firft delivered by 
Mr. Berznoutti, Mr. De Moivre, Mr. Evier, &c. 
12. Affume the feries a4 + bx” 4- cx?” + &c. =f, multiply it into 
wx, and find the fluent, thei will : up — : fv = : axe + 
| Ggeg2 f 
