Ill 
Infinite Series . 
greateft roots of the equations before-mentioned be greater than 
the unknown quantities ; and unlefs the unknown quantities 
have a great ratio to the greateft roots the feriefes will converge 
(lowly : for example, the feriefes^ — i x z + &cc. 9 J* 
= z dL==r==y q--iy 3 +& c * will never con- 
verge if x 9 z, or y 9 be greater than i ; but will always con- 
verge when lefsthan ±r or ±i V-i theleaft or only roots of 
the equations i i —y 2 = o, and i-|-z 2 = o. The feries 
. j-& c . will always converge when y is fituated between 
+' I and — i, in which cafe alone it is poffible. The fame is 
true alfo of a feries arifing from expanding the f (ax m -f bx m ~' + 
ex™-" 2 + &c.y m x into a feries proceeding according to the dimen- 
fions of x 9 if the equation ax* 4- bx"~ x + cx m ~ z -f &c. — o has 
only two poffible roots a and — a, which are Ids in the manner 
before-mentioned than any impoffible root contained in it. 
If in either of the above-mentioned feriefes the unknown 
quantity x 9 z 9 or y 9 has a great proportion to i, the 
feries will converge very flow ; for example, if x—i, ten 
thoufand numbers at leaft are to be calculated, to procure 
the fum of the feries true to four figures ; therefore, in thefe 
and moft other feriefes it is neceftary fir ft to find a near value, 
viz. when x either = z, when e is very final 1 ; or— e r when z is 
very final! * and then write z + e for x in the quantity, and 
reduce it in the former cafe into a feries proceeding according 
to the dimenfions of in the latter cafe according to 
the dimenfions of z, and there will arife two feriefes, of 
which the fluents properly corrected, viz. by adding the 
fluent contained between the values a and e to the latter, 
and that between a and z to the former, will give the .fame: 
fluent- 
