RULE.

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

into an infinite feries.

6. To throw

2 2x2

-X into an infinite feries.

1+x -3x

Ans. 2xa—2x+7x3 — 13x2+34x3 &c.

PROBLEM II.

To reduce a compound furd into an infinite feries.

RULE.

Extract the root as in common arithmetic, and the operation, continued as far as fhall be thought neceffary, will give the feries required.

EXAMPLES:

1. Extract the fquare root of a2+x2 in an infi

nite feries.

3. Throw a'-x into an infinite ferics.

To reduce a binomial furd into an infinite feries; or to extract any root of a binomial.

Subftitute the particular letters of the binomial, with their proper figns, in the following general form, and it will give the root required; obferving that P is the firft term, Q the fecond term divided

by the first, the index of the power or root; and

12

A, B, C, D, &c. the foregoing terms, with their figns.

P+PQ"=P" (A) + "AQ(B)+”=”BQ (C) +

n

m—2nCQ (D)+”=3" DQ (E) &c.

372

4n

EXAMPLES:

1. To extract the fquare root of 2x2, in an

infinite feries.

422

-B + C+ -D &c.

622

And, by reftoring the values of A, B, C, D, &c.

&c. feries required.

2. To extract the cube root of

finite feries.

This expreffion, reduced, is

a2+x2