114

ON CUBIC EQUATIONS

TRACT 28.

118. Again,

K - y A

TT

C =

= -3333333

330

C = fB =

& — TT D =

+ -3333663- -0020413

•0020406

7

•0020413

•3313250

2

log. 1-8212842

•6626500 - - -

■v/3 ---

£/121 ---

the latter series ± -2320508 ----- 1-3655830

A the first — 2-2320508

- - 0-2385606

- - 0-69426)8

middle root =— 2-4641016least root =— 20000000

119. But, by Art. 64, the three roots are —2 and l±^/12;hence

l_ + 2^321/112-^3

= 1 +

2. S. 8.2*

2.5.8 . 11 . 14.2®

3.6. 11 2 3 . 6.9 . 12 . II 4 ' 3.6. 9 . 12.15

2.5.8.11.2^

376.9 ril’ J

3/121 = 1 --(-

V o <j <; o m 1

nor. 4c »

_ &c

4 v - 3 3.6.9.IP ' 3 . 6.9 . 12 . 15 . II 4

120. And in this manner the roots of cubic equationsmay always be found by these series; and then, by compar-ing them with the roots of the same equations, as found byother methods, we shall obtain as many series as we please,whose sums will be given.

121. Hence also we may find the sum of any generalseries of either of these forms, namely,

2.5. 8g4 2.5.8.11. 14g s

3.6 3767 9 . 12 —

2.5g 2 2.5.8. llg*

1

S’-k 3.6.9

+ 5

3.6.9 . 12.15.18. 2 . 5 . 8 . 11 . 14.17ff«

&c, or

&C, by

3.6 . 9.12 . 15 3 . 6 . 9 . 12.15 . 18.21

comparing them with the roots of given cubic equations;whatever be the value of g, not greater than 1.

122. For, by Art. 68, l/{b + c ) +\/{b — c)—- 2 %/b x : 1 —2c 2. 5 . 8c4 ^ greatest root of the cubic

3.66 J

3 .6.9. 126<