Summation of Series. 391 
-/This method may be extended to infinite feries, in which 
exponentials as e* are contained, which will eafily be feen from. 
fome fubfequent propofitions ; but in my opinion the fubfequent 
méthod of finding the fum of feriefes is tobe preferred to the 
preceding one, both for its generality and facility. 
6. 1. Let the general term be (ax? + bzP—? 4 xh? 4 Bc. De 
(ste)? . (2+e+1)7*. (@@tep2)—. . (84+e42-1)75 
where 4 is a whole number lefs than x a two or more, when 
the fum of an infinite feries is required. 
Affume for the fum the quantity (s+e)—. (2+e+ ry cene 
(Sfpep2) (sheet a = 2) y (az? + Bh yh yelp &c.); 
find the difference between this fum and. its fucceflive one 
(wet Ori AS AVS! Cz ted 3)5" ae (s+etna— Tae 
ise (ezt1 I Da. ati | Sos &c.), which will be —(z4+¢e)—. 
Geet BSE Ae UE ating Sida ea a, 
(as+1 I agate Pam a I teas -- S&C. ) ae a Cams 
(=h —an+ tex + &e Ni; their make the terms, of this dit 
ference equal to the correfpondent terms of the given quantity 
az? + bz 4 &c. and there refult 2’ =h,—-h-n+1xa=a, and 
Dyas ‘i 
confequently «= a or 5 Bae 
22, Let “the -veneral term ‘be « (e-fe)"* 2 @ eden) 
(sfet2y—7...(2fet+u-1)7x@t+f/)7.@+ft+ i pag 
(24 f+2)*... (2+ f+m—1)— x (az? + bz) 4 cz’* 4 &e.). 
Affume the quantity (= +e)". (z+e+1)—...@@+e+n—2)> 
x(2+f)—.. (2t+f+i)7 . (@4+f+2)".. (z+ f4+m-—2)7 
x (ax +622 472° 4+ &c.) for the fa of the feries fought ; 
and thence deduce the general term, which fuppofe equal to the 
given general term, and from equating their correfponding parts 
eafily can be deduced the index 4’ and co-efficients a, @, y, &e:. 
and confequently the fum of the feries fought.. 
a. eee 
