491. If the series be y = 1 + 2x + 3x^2 + 4x^3 + ... put y - 1 = z;
then z = 2x+3x^2 + 4x^3 + .
Assume x= Az + Bz^2 + Cz^3 + and complete the work as in Art. 490 after which replace z by its value y-1.
EXAMPLES XLII. d.
Revert each of the following aeries to four terms :
1. y = x + x_2 + x_3 + x_4 + 2. y = x + 3x_2 + 5x_3 + 7x_4 + 3. y = x - x_2 2 + x_3 4 - x_4 8+ 4. y = 1 + x + x_2 2 + x_3 6+ x_4 24 + 5. y = x - x_2 3 + x_3 5 - x_4 7+ 6. y = ax + bx_2 + cx_3 + dx_4 +.
PARTIAL FRACTIONS.
492. A group of fractions connected by the signs of addition and subtraction is reduced to a more simple form by being collected into one single fraction whose denominator is the lowest common denominator of the given fractions. But the converse process of separating a fraction into a group of simpler, or partial, fractions is often required. For example, if we wish to expand {3-5x}{1-4x+3x^2} in a series of ascending powers of x, we might use the method of Art. 487, and so obtain as many terms as we please. But if we wish to find the general term of the series, this method is inapplicable, and it is simpler to express the given fraction in the equivalent form {1}{1-x} + {2}{1-3x} Each of the expressions (1-x)^-1 and (1-3x)^-1 can now be expanded by the Binomial Theorem, and the general term obtained.
493. We shall now give some examples illustrating the decomposition of a rational fraction into partial fractions. For a fuller discussion of the subject the reader is referred to treatises on Higher Algebra, or the Integral Calculus.
