Where the law of formation of the several terms

of the series is sufficiently evident.

(a

2. It is required to find the value of (+6) its equal (a+b)-2, in an infinite series.

b

m

or

Here Pα, a=- and -=- 2, or m=-2 and

And if the reciprocals of the same expressions be required,

3. It is required to find the value of

its equal a (a2 - x) ̄*, in an infinite series.

Here

564

a2

(a2 - x)

or

It may here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative, the first of these expressions may be exhibited in the more simple form

Where the last term is called the general term of the series, because if 1, 2, 3, 4, &c. be substituted successively for n, it will give all the rest.

4. It is required to find the value of (8 + 1)3,

/9, in an infinite series.

or

=

B,

1 1

1

2n

6

3.22

3.6.24

= C2

1

2+

1

3.22 3.6.24 3.6.9.27 3.69.12.20 +3.6.9.12.15.213&C.

5. It is required to find the value of √2, or (1+1), in an infinite series.

6. It is required to find the value of (8-1)*,

or 7, in an infinite series.

7. It is required to find the value of (a±x)*,

in an infinite series.

Ans. a{1+

x

x2

3x3

3.5x4

+

2a 2.4a2 2.4.6a2 2.4.6.81 ± &c.}

8. It is required to find the value of (a±b)*,

in an infinite series.

Ans, a3 {1+

b

262 2.563 2.5.864
+

3 a 3.6a2 3.6.9a3 3.6.9.120&c.}

9. It is required to find the value of (a−b)*,

in an infinite series.

Ans. a+1

3.7.1164

&c.}

b 362 3.763
4a 4.8a2 4.8.12a3 4.8.12.16a4

10. It is required to find the value of (a+x)*,

in an infinite series.

Ans. a{1+

2x x2 4x3 4.7x4 4.7.10x5

+

+

3a 9a 92a3 9212a4 92.12.15as

11. It is required to find the value of

24 2.4a2 2.4.6a3 2.4.6.8a4

&c.}

1

in

Ans. 17 +

干

+

F&c.}

2x2 2.5x3
3a 3.6a2 3.6.9a3 3.6.9.12a

13. It is required to find the value of (1 − x)

in an infinite series.

Ans. 1

2x 2.3x2 2.3.8.x 2.3.8.13.x4
5 5.10 5.10.15 5.10.15.20

14. It is required to find the value of

x

15. It is required to find the value of (+), in

an infinite series.

x3 3x4 3x5 5x6 5x7

Ans. 1++ + + + + +

a 2a2 2a3 8a4 8a5 1646 16a79

&c.

OF THE MULTIPLICATION AND DIVISION OF SERIES.

(z) ANY two series may be multiplied or divided by each other, by substituting the numbers, or letters, of the given example, in the place of their corresponding terms, in the following general formulæ, and then finding the result accordingly (k). Â。 + Â ̧x + А„x2 + Â ̧Ã3 +A4x1&c.

B。 +  ̧Ã+  ̧Ã2 + „Ã3 +  ̧Ã1&c.

And if B. A, B, A,, &c. the formula for the

square of the

=

upper, or following, series,

Ao + A1x +Аx2 + A3X3 + A4X+&c.

will, in that case, be as follows:

(k) In the mode of notation here followed, it is to be observed, that A, A, A,, are the coefficients of the several powers of x, so taken, that the inferior figures 0, 1, 2, 3, &c. by which they are denoted, shall always agree with the indices of the unknown quantities that are attached to them; by which means, the law of the continuation of the terms is, in all cases, rendered obvious, and a conformity between the coefficients and powers of x constantly preserved.