TRACT 28.

AND INFINITE SERIES,

101

And to this belongs all cases of the equation that can fallin between the 13th and 16th formal®, in the general tablein Art. 30.

79. If a 3 be == fr, then c ~O y and the three first seriesgive 2 %/b — l/iq for the greatest root, and — l/b for each ofthe less roots. The same as at the 13th form in the generaltable Art. 30.

80. When a 3 is greater than b z , er will be negative, andthen, changing-the signs of the odd powers of c% the threegeneral series will give the three roots of the equation,which will always be all real. In this class are two cases,namely, when (f is Jess than 6% and when they are equal,which is the limit; for when c % becomes greater than 5% theseries diverge.

81. Now when a 3 is between and 2^, then r is negativeand less than b z , and the general series give all the three realroots, by changing the sign of every other term.

82 1 . And when a 3 = 2& ? , then -c i = b z , and the threeroots become thus:

first or greatest root,

the two less roots.

83. The first of these three is the greatest root, because

So that in general the first series gives the greatest of thethree roots.

84. But it is evident, that this case agrees ivith the 10thform in the table Art. 30; in which the middle root r is

-1/b x : 1 4-

±l/b x V3 x

2.5, 3

2 . S ■ 8 . II . U

.&C,

and

3.6.9. 12

3 . S . 9 . 12 15 . IS

£. 5S . 6. 9

+

3.6.9 . 12 . 15

f/b x : 1 + — —^t 4 5 * * n ’ 8 -- &c, is greater than fyb x x:

3 • 6 3 • v , 9 .

4- — Stcg/for 1 &c, is greater than 1, and

>5 3«d.9 3.0

i/Sx : — - 3 ^ 6 ~ &c, — v'-l- X : 1 — |^| &c, is less than 1.