&c. 
of Infinite Series . 303 
-4-2 hyp. log. 2 ; and by Prop. 1. A 4 C 4- E 4- &c. ~ 
5 ; hence b + d+f+&cc. = - - +2 hyp. log. 2. 
4 4 
PROP. IV. 
To find the fum of the infinite f cries whofe general term is 
mx zh n 
By divifion 
4. — 
mx ziz.n mx 
r — s z zr . — s o 7r — - $ 
m x m s x 3 
&C. 
ad inf \ ; 
hence the fum of the feries of which 
mx ±~n 
is the general term, 
is found as in Prop. 2. Here r mu ft be greater than s at leafi 
by 2, otherwife the fum will be infinite. 
Ex. 1 • Let the general term be -f— = — 2 — 4- 1 — &c. ; 
0 A" 4* x x l ° 
hence if for * vve write 2, 2, 4, &c. we have — + -^4.^!. 4. 
J i 7 T 8* T a5 7 ■ 
&c. = A- E + I — N+ &c. =,538527924723. — If for x we 
write 2, 4, 6, &c. we get 4 _U— +4: _i_ &c. = A." - E" + 1 
T 0 17 1 257 1296 1 
-N" + &c. = ,396257616555. 
Ex. 2. Let the general term be — - — = — 4. JL + _i_ 
0 3 ^ — 1 3 # z 9^5 27^ 
// 
4*&c. 
hence if we write 2, q. 4, &c. for 4 we have — 4 4- — 4- &c. = 
J 23 80 191 
~ A 4* ~D4--^G+&c. =,219238483448* 
By this propofition we may find the fum of any feries whofe 
general term is — - hx -■ ■ *** — ; for this refolves itfelf into 
lx*~ T 
mx zzn 
ax 
r r 1 
mx ztn mx ±:u 
6 
&c. &C., the fum of each of which feries is 
found 
