Elements of Algebra

CHAP. VI. OF ALGEBRA. 97 markable, that an infinite series, though it never ceases, may have a determinate value. It should likewise be observed, that, from this branch of mathematics, inventions of the utmost importance have been derived; on which account the subject deserves to be studied with the greatest attention. QUESTIONS FOR PRACTICE. 1. Resolve $ \frac{ax}{a-x} $ into an infinite series. Ans. $ x + \frac{x^2}{a} + \frac{x^3}{a^2} + \frac{x^4}{a^3}, \&c. $ 2. Resolve $ \frac{b}{a+x} $ into an infinite series. Ans. $ \frac{b}{a} \times (1 - \frac{x}{a} + \frac{x^2}{a^2} - \frac{x^3}{a^3} + , \&c.) $ 3. Resolve $ \frac{a^2}{x+b} $ into an infinite series. Ans. $ \frac{a^2}{x} \times (1 - \frac{b}{x} + \frac{b^2}{x^2} - \frac{b^3}{x^3} + , \&c.) $ 4. Resolve $ \frac{1+x}{1-x} $ into an infinite series. Ans. $ 1 + 2x + 2x^2 + 2x^3 + 2x^4, \&c. $ 5. Resolve $ \frac{a^2}{(a+x)^2} $ into an infinite series. Ans. $ 1 - \frac{2x}{a} + \frac{3x^2}{a^2} - \frac{4x^3}{a^3}, \&c. $ CHAP. VI. Of the Squares of Compound Quantities. 306. When it is required to find the square of a compound quantity, we have only to multiply it by itself, and the product will be the square required. For example, the square of $ a + b $ is found in the following manner: H