TRACT 2S.

AND INFINITE SERIES.

10S

where the signs of the former series are found by changing-the signs of every other pair of terms in the latter; namely,omitting the first term, change the signs of the 2d and 3dterms, then passing over the 4th and 5th terms, change thesigns of the 6th and 7th ; and so on. For, by Art. 86, the

former of these series is equal to ~ ; and, bjr Art. 72, thelatter is equal to^/2.

90. Let us now consider the cases in which c z is greaterthan b z , which include all the cases not comprehended bythe former, or in which c 2 is not greater than b And this,it is evident, will happen both when a is positive, and whennegative; namely when a is any positive quantity what-ever, or when it is any negative quantity, and a 3 greaterthan 2 b z . And in these two classes, & will be positive ornegative, according as a is positive or negative.

91. Now the series in this class will be found the sameway as in the last, by only writing here the letter c beforethe letter b ; for then we shall have s =^{c -f b ), and d —

c + b) = — */(c - b).

Then s —^/{c + b) =*/e x : 1 + -

3.6

, 2 . 563

<.> + 3 . 6 . y t 3 &c ’

and d — -?/{c — b) —l/cx 1 +j- + y-r-; +

Hence s + d =

2 b

3 c

1 2.54»

X ’ s' 3.6 . 9c=

+

3.6 c* 1 3.6.9 c32.5 . S . Ill*

&C.

' 3/c 2 ~ ’ 3 1 3.6 . 9c« 1 ' 3.6 . 9 . 12 . 15c 4the 1st root, and which was given by Clairaut. And

&c

d

& c

-v' —3

-b w _ 1 , 2 . 5b* , 2 . 5 . 8 . 11M

JV? X ' 3 T" 3.6.9? 3.6. 9 ,7a . 15<*

I , .. . „ , 24= 2.5. 84 4 0

±.^. ty - 3 X : 1 - —- r - T7T _ &c

for the other two roots, which are new.

92. Here it again appears, that when c z is positive, thetwo latter roots are imaginary; because then 3 /c x \/ — 3will be imaginary. But if c z be negative, those roots willbe both real; since %/c x s / — 3 then becomes %/(c V — 1) xV ~3 —%/c x - y'-lx V' — 3 = — l/c x \/ 3. The signs pre-fixed to the terms as above, take place when c 1 is positive ;but when c z shall be negative, the signs of the terms con-