EXAMPLES.

1. It is required to find the square root of 9/3.

1

1

=36; =

First, √9=3, and (33)3⁄4—33÷2—38;
Whence, (93)=3.38 square root required.

2. It is required to find the cube root of 2.

1

3

First, ✅- , and (√2)31=231÷3—2';

8

2

1 1 2

Whence, (=—= √2) 3 =- -26 =cube root required.

3. Required the square root of 103.

1

Ans. 10/10.

6. It is required to find the nth root of xm.
7. Required the square root of x2—4x√✅a+4n.

INFINITE SERIES.

An infinite series is formed from a vulgar fraction, having a compound denominator, or by extracting the root of a surd quantity; and is such, as, being continued, would run on ad infinitum, in the manner of a decimal fraction.

But, by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued, without actually performing the whole operation.

>

1

PROBLEM I.

To reduce fractional quantities into infinite series.

RULE.

Divide the numerator by the denominator, as in common division; and the operation continued, as far as may be thought necessary, will give the series required.

1

2. Let

1+x

be converted into an infinite series.

1+x)1.....(1-x+x2-x3, &c.

be converted into an infinite series.

Ans. 1+2x+2x2+2x3+2x4, &c.

be converted into an infinite series..

(a+x) 2

PROBLEM II.

To reduce a compound surd into an infinite series.

.RULE*.

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

EXAMPLES.

1. Extract the square root of a2+x2 in an infinite

This rule is chiefly of use in extracting the square root, the

operation being too tedious for the higher powers.

2. Let 1+1 be converted into an infinite series.

3. Let a2x2 be converted into an infinite series.

4. Let 1-3 be converted into an infinite series.

5. Let a+b be converted into an infinite series.

PROBLEM III.

To reduce a binomial surd into an infinite series; or to. extract any root of a binomial.

RULE*.

Substitute the particular letters of the binomial with their proper signs, in the following general form, and it will give the root required; observing that P is the first term, a the second term divided by the first, m the index of the power or root; and A, B, C, D,

n

foregoing terms with their proper signs:

P+QP¬=P¬ (A) +

m

&c. the

m

m

AQ (B) +

BQ (c) +

2n

3n

* Any surd may be taken from the denominator of a fraction and put in the numerator, and vice versa, by only changing the sign of its index.