6. It is required to convert the rational fraction

1 + x2 x-x3

into its equivalent simple fractions.

Here the factors of the denominator x-x being x, 1-x, and 1+x, we shall have

as in example 2; whence, for x, the first factor,

A =

N 1 + x2

S

=1=1, a being =0; and for 1-x, the

simple fractions, as before.

7. It is required to convert the rational fraction

1

x3( 1 − x)2(1 + x)

into its equivalent simple fractions.

Here, the factors of the denominator being a, (1-x), and 1+x, we shall have

N

whence, for 1+x, the first factor, A=-=

1

1; and for x, the second factor,

=1, x being 0.

N-BS x + x2-x3

N

B=

S

x3

Let now a=

=1+x-x2;

then, for x,

2-x2

B′′===—=2, x being =0.

Also, for (1 − x)°, c ====, being

x = 1.

1

S

+ 204 2'

OF RECURRING SERIES. A

(k) A RECURRING SERIES is a rank, or progression of quantities, so constituted, that each term has a constant relation to some given number of the preceding terms, taken continually in the same order.

Thus, if 1+6x + 12x2 + 48x3 +120x* &c. be the given series, we shall have the 3d term 12x=x× 2d term + 6x x 1st term; the 4th term 48x3 = x× 3d term + 6x2 x 2d term; and so on.

In which case, the compound expression x, 6x, or simply 1, 6, is called the scale of relation of the several terms.

And if 1+4x+6x2 + 11x3 +28x+63x5 &c. be the given equation, we shall have the 4th term 11x3 2xx 3d term - x2 x 2d term + 3 x 1st term; the 5th term 28x*= 2x × 4th term - xx 3d term +3xx 2d term; and so on.

2

Where 2x, x2, +3x3, or 2, −1, +3, is the scale of relation; the numbers composing it being the multipliers by which the several terms of the series, or their coefficients, are produced (s).

(s) It

may be here observed, that series of this kind arise from the development of certain fractional expressions, of the form

in which the terms of each denominator, abating unity, constitute the scale of relation of the terms of the series, produced from the expansion of the fraction in question.

Thus, if A, B, C, &c. be made to denote the penultimate co

PROBLEM.

Having given an infinite recurring series, of the form a + bx + cx2 + dx3 + ex &c. of which the scale of relation of the terms is known, to find its sum.

RULE.

1. When the scale of relation consists only of two terms, as +a', +b', the radix, or sum (s) of the series, infinitely continued, will be

2. When the scale of relation consists of three a', +b', +c', the sum of the series, in

terms,

finitely continued, will be

S=

a + (b-aa')x+(c-ba' — ab') x2

1-a'x - b'x2 — c′x3

3. In like manner, when the scale of relation

efficients of the successive terms of the series arising from the development of the first of these fractions, we shall have

Where it is evident, that the coefficient of each term is formed by multiplying that immediately preceding it by a', or the entire term by a'x; in which case, therefore, -a', or -a'x, is the scale of relation of the several terms of the series.

Again, if the second of these fractions be converted, in like manner, into a series, we shall have

= a + (b + a's)x — (b′a + a'ï)x2 — (b'B + a′c)x3 →→ b'x

(b'c + a'D)x+ &c.

Where each coefficient, beginning with the third, is determined by means of the two that precede it, multiplied respectively by

consists of four terms, +a', +b', +c', +ď, the sum of the series, infinitely continued, will be

a + (b−aa')x + (c− ba' — ab')x2 + (d — ca' — bb' — ac'}x3

1-a'x-b'x2 — c'x3 — d'x+

And so on, for any greater number of terms, observing to change the signs of the quantities a', b', c′, &c. which are here supposed positive, from - to, when any of them are negative (†).

EXAMPLES.

1. Required the sum of the infinite recurring series 1+6x+ 12x2 + 48x3 + 120x &c. the scale of relation of its coefficients being 1, 6.

Here a=1, b=6, a′ = 1, and b′=6; whence,

` — b', — a'; or each entire term, by the two preceding terms, multiplied by b'x, a'x; the quantities-ba', or b'x, - a'x, constituting, in this case, the scale of relation of the terms of the series.

And a similar mode of proceeding may be employed in the development of either of the other fractions; in which cases, however, as well as in those given above, the operation may be more perspicuously performed by the method of indeterminate coefficients, used in Art. c.

(t) It is sometimes difficult to determine whether a given series be recurrent or not; in which case, the following rule, first given for that purpose by Lagrange, in the Mem. de l'Acad. des Sciences de Paris, année 1772, will be found useful.

Divide unity, or 1, by the sum of the proposed series, s, as far