Infinite Berks. io? 
— ~r^-+ fjL + rj±+ &cc.=(-+ 1 + 
J a+tx+tx 7, . . x" J « — * J &—* J y—* '* ® 
- +&c.) x + (\ +1 + t+ &c.) y 2 + &c. always converges 
when ^ is between & and — where ^ is the lead: root of 
the above-mentioned equation ; but where x is greater than a 
or - u 9 the feries will diverge. 
The infinite feries a m + ma m ~~ l x 4- m . a™— 1 x* + &c. rr a+x 
will always converge when a is greater than and diverge when 
lefs, and confequently its convergency does not depend on the 
index m 9 utilefs when x~~a: and in the Mcdltationes Analy- 
tics are given the cafes in which it converges or diverges when 
qztf — x; andalfo if the feries x m + &c. ~x + a defcends 
according to the dimenfions of a:, when it converges or diverges. 
Caf. 2. Let the above-mentioned quantity be reduced into a 
feries Ax~ r 1 +&c. defcending according to the dimen- 
fions of the unknown quantity x ; then will the feries Ax~ r -\- 
B v — r— * _j_ g lCt — p or the feries Ax - — — - -p &c. = f Pi con- 
I — r r J 
verge, when x is greater than the greatefl: root (a) of the 
above-mentioned equations, and diverge when it is lefs ; and 
confequently in this cafe, when the fluent is required between 
the two values a and b of x; the feries found will converge 
when a and b are both greater than A. 
Caf. 3, When x is equal to the lead: root in the former cafe, 
and to the greatefl: in the latter, then fo me times the feries will 
converge, and fometimes not. Thefe different cafes are given 
in the Me dit at tones ; but it would be too long to infert them in 
this Paper. 
4. If any roots are impoffible as a-bs/ - 1 and a-\-bs/ - 1, 
then the feries will converge when x is in the firfl: cafe 
Vol.LXXVI. ° P lefs 
