INFINITE SERIES. AN Infinite Series is formed either from division, dividing by a compound divisor, or by extracting the root of a compound surd quantity; and is such as, being continued, would run on infinitely, in the manner of a continued decimal frac tion. But by obtaining a few of the first terms, the law of the progression will be manifest; so that the series may thence be continued, without actually performing the whole operation. PROBLEM I. To Reduce Fractional Quantities into Infinite Series by Division. DIVIDE the numerator by the denominator, as in common division; then the operation, continued as far as may be thought necessary, will give the infinite series required. To Reduce a Compound Surd into an Infinite Series. EXTRACT the root as in common arithmetic; then the operation, continued as far as may be thought necessary, will give the series required. But this method is chiefly of use in extracting the square root, the operation being too tedious for the higher powers. EXAMPLES. EXAMPLES. 1. Extract the root of a2x2 in an infinite series. a3 - x2 (α- 20 δαν 5x8 &c. 2. Expand 1+1=√2, into an infinite series. Ans. 1+-+/- &c. 3. Expand 1-1 into an infinite series. £ 128 Ans. 1-1-1-/- &c. 4. Expandax into an infinite series.. PROBLEM III. To Extract any Root of a Binomial: or to Reduce a Binomial Surd into an Infinite Series. THIS will be done by substituting the particular letters of the binomial, with their proper signs, in the following general theorem or formula, viz. m 2n 3n and m and it give will the root required: observing that P denotes the first term, the second term divided by the first, the index of the power or root; and A, B, C. D. &c., denote the several foregoing terms with their proper signs. EXAMPLES. m m 1. To extract the sq root of a2+b2, in an infinite series. (a2)=(a2)=a=A, the 1st term of the series. 62 b2 1-2 b2 h2 B, the 2d term. =c, the 3d term. D the 4th. 64 3.66 Hence a+ + -&c. or 2.4.6a5 *Note. To facilitate the application of the rule to fractional examples, it is proper to observe, that any surd may be taken from the denominator of a fraction and placed in the numerator, and vice versa, by only changing the sign of is index. Thus, m-2n 3x2 -X 4x3 X. 4a 5x3D. 3n a4 α αδ Hence a ̄ +2α ̄3x+3α ̄ ̄1x2+4a ̄ ̄x3+ &c. or 1 2x 3x2 4x3 5x4 + + + +. a2 a3 a4 a5 &c. is the series required. ασ 6. To expand ✔a2 — x2 or (a2 —x2)1⁄2 in a series. xa х Ans. a 2a 8a3 16a5 128a7 7. Find the value of 3⁄4/(a3—b3) or (a3 — b3)3 in a series. 8. To find the value of £/(a*+x3) or (a3+x3) in a series, |