Here it is easy to see how the succeeding terms of the quotient may be obtained without any further division. This law of the series being discovered, the series may be continued to any required extent by the application of it.

I.

1+x

4

=-x+x*x3+x*-, &c. the answer.

Here the exponent of x also increases continually by r from the second term of the quotient; but the signs of the terms are alternately + and

3. Reduce

+x)(++, &c. and

C

a

C

a+x

is the answer.

* Here we divide c by a, the first term of the divisor, and

the quotient is, by which we multiply a+x, the whole divi

ac

a

cx

CX

sor, and the product is -+- or c+, which being subtract

a

ed from the dividend c, there remains

a

CX

ing divided by a, the first term of the divisor, gives

329

term of the quotient, by which we also multiply a+x, the divi

The rest of the quotient is found in the same manner; and four terms being obtained, as above, the law of continuation becomes obvious; but a few of the first terms of the series are generally near enough the truth for most purposes.

And in order to have a true series, the greatest term of the dis visor, and of the dividend, if it consist of more than one term, must always stand first.

Thus in the last example ; if x be greater than a, then & must

be the first term of the divisor, and the quotient will be

arc + x3

a3c
4

C

x+a

+, &c. the true series; but if x be less

than a, then this series is false, and the further it is continued, the more it will diverge from the truth.

For let a=2, c=1 and x=1 ; then if the division be perform

ed with a, as the first term of the divisor, you will have

But if x be placed first in the divisor, then will

I

1+2=1-2+4-8+16-, &c.

C

a+x

Now it is obvious, that the first series continually converges to the truth; for the first term thereof, viz., exceeds the truth by

!

:

or; two terms are deficient by; three terms exceed it by; four terms are deficient by ; five terms will exceed the truth by , &c. So that each succeeding term of the series brings the quotient continually nearer and nearer to the truth by one half of its last preceding difference; and consequently the series will approximate to the truth nearer than any assigned number or quantity whatever; and it will converge so much the swifter, as the divisor is greater than the dividend.

But the second series perpetually diverges from the truth; for the first term of the quotient exceeds the truth by I, or two terms thereof are deficient by ; three terms exceed it by ; four terms are deficient by; five terms exceed the truth by , &c. which shew the absurdity of this series. For the same reason, x must be less than unity in the second example ; if x were there equal to unity, then the quotient would be alternately 1, and nothing, instead of; and it is evident, that x is less than unity in the first example, otherwise the quotient would not have been affirmative; for if x be greater than unity, then 1-x, the divisor, is negative, and unlike signs in division give negative quo sients. From the whole of which it appears, that the greatest term of the divisor must always stand first.

:

PROBLEM II.

To reduce a compound surd to an infinite series.

RULE.*

Extract the root as in Arithmetic, and the operation, continued as far as may be thought necessary, will give the series required.

EXAMPLES.

2

1. Required the square root of a2+x* in an infinite

* This rule is chiefly of use in extracting the square root; the operation being too tedious, when it is applied to the higher powers.

† Here the square root of the first term, a2, is a, the first term of the root, which, being squared and taken from the given surd a2+x2, leaves ** ; this remainder, divided by 2a, twice

2

2

the