: ‘ | ; 
394 Dr. Wanine on the , | 
Some other writers have made fome feriefes to appear more 
difficult to be fummed, by not reducing them to their loweft : 
terms. 4 
+, Having given the principles of a general method of find- 
ing the fum of a feries, when its general term can be exprefled 
by algebraical, and not exponential, functions of z, the dif- 
tance from the firft term of the feries; it remains to perform 
the fame when exponentials are inchided. : 
1. Let S the fum be any algebraical function of 3 multi- 
plied into e* =x* ; then will the general term be Se*7—eS’e#= 
(S— eS’); whence, from the general term Te* being given, 
affume quantities in the fame manher (with the fame denomi- 
nator, &c.) as when no exponential was involved, which 
multiplied into e%, fuppofe to be the fum; from the fum find 
its general term, and equate it to the given one ‘by equating © 
their correfpondent co-efficients, and it is done. 
Z+2 
227-1. 2343 
Ex. Let the general term be x e=tr: affume for.’ 

the fum fought x et1, whence the general ‘term is 
x 
2z24+1 
r \e ary) 
( ra om ) eases ae 



x e*+1$ equate it to the 
22-1 23 -+ 3 25-1... 2243 
given term, and there refults 22 (1—-¢)=1 and 3a— oe = 2, 
and confequently e=+ and a=, if the feries can be fummed. 
The fame obfervation, vz. that if any fa€tor in the deno= 
minator or irrational quantity have no other correfpondent to 
it; for example, if the faCtor be 24+, and there as no ¢or-" 
re{pondent one ¥+g+%, where n is a Whole number, then its 
integral cannot be exprefled by a finite algebraical fun@ion 
of z. | 
In the fame manner may the fums be found, when the terms 
are exponentials of fuperior orders; for the exponential, inra-. 
6 tional, 
