Summation of Series. 389 
where 4 is a whole number lefs than 2+m4+/+4&c. (if it be 
ereater, then the fra¢tion can eafily be reduced into a rational 
quantity ax’—"—"—'—** + &c, and a fraction of the'before-men- 
a al 
tioned kind); then VU gh = (ae te + &c.) + 

8 ea 6” 
(gary ~-2+et+i1 uss a Page pagel rita i Zte+t + &c.) + 
Piaaeie OT is O05) Coty fr i eae ny 1 | 
E> Saag tf. Btft+i.ztf+2 2tg.z+g+1.2+e+2 
: 9 F Xo ; xy’ 
&c. eee (ee EMOTR ay ekg NUE Ry os VO 
ik ) aes Give RE ef see ie tee 
: ay ‘ / % 4 
pr f iS 2 Ww H . . i 
emia ~2t+g+1..2+g+/—1 +.&c ) 2 whence it¢ integral 1n tn 
fuitum,,that is, the.fum of the infinite feries can be found. 
melicn @= 0.4 =0, (eo = 0, c.; and confequently Ab not 
greater than #+- +/+&c.—2; otherwife-not. If 4 is not 
greater than 7 -+m +/+ &c. —2, then willa te’ +a” +&c.=0,. 
for elfe the fum would be infinite. 
Let the number of quantities (¢, /, g, &c.) ber, then from . 
y independent. integrals of a feries, whofe term _is_’I’; or 
from (r —1) independent {urns of infinite feriefes, whofe term 
is I”; that is, where / is not greater than n+m+/4+&c. — 23 
can be deduced the fum of all infinite feriefes of the before- - 
mentioned formula, whofe general term isv EL’. 
If any factors are deficient in the denominator, as fuppofe the 
term to be stexz+e+3x2%+e+2-—1; multiply the nu- 
merator and denominator by the deficient factors, viz. by 
epeta.ztetaxetet+q).sbepo..stetn—2, andit 
acquires the preceding formula; and fo in:the following 
examples. 
3d, Let the denominator be x + o x oa eae iy 
ee BSG Le BO Ol ES Ng EN Seen? 
Met? «6 KE OCPA KE AWE eK ee wed. x: 
#*+¢é 
