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.

2. Expand 1+1=2, into an infinite series.

Ans. 1 + − + − ‚1⁄2 &c.

123

3. Expand 1

1 into an infinite series.

Ans. 116-118&C.

4. Expanda2 + x into an infinite series.

5. Expand va2-2bx-x2 to 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.

2n

BQ+

cQ+ &c.

3n

and

and it will give 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.

1. To extract the sq. root of a2 + b2, in an infinite series.

≈≈ B, the 2d term.

a2 2a

2.4a3 a2 2.4.6as

2.4a3

=D the 4th.

b2

64

3.66

Hence a +

2.4.6as

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 its index. Thus,

(av = x2)! = (a° + x3) 1 × (aa—x2) ̃ ̄2; &c.

Here

Hence a2+2a ̄3x + 3a2x2 + 4a ̄3x3 + &c, or

6. To expand √a2 — x2 or (a2 — x2) in a series.

Ans. a

x2

5.x8

&c.

2a 8a3 16as 128a7

I

7. Find the value of 3⁄4/(a3 — b3) or (a3 — b3)3 in a series.

63 bo 569

&c.

8. To find the value of 5/(as+xs) or (as +xs) in a series.

2x10 6.xls
+

&c.

25a 125a14

ARITHMETICAL PROPORTION is the relation between two numbers with respect to their difference.

Four quantities are in Arithmetical Proportion, when the difference between the first and second is equal to the difference between the third and fourth. Thus, 4, 6, 7, 9, and a, a +d, b, b+d, are in arithmetical proportion.

Arithmetical Progression is when a series of quantities have all the same common difference, or when they either increase or decrease by the same common difference. Thus, 2, 4, 6, 8, 10, 12, &c, are in arithmetical progression, having the common difference 2; and a, a + d, a + 2d, a +3d, a + 4d, a + 5d, &c, are series in arithmetical progression, the common difference being d.

The most useful part of arithmetical proportion is contained in the following theorems :

1. When four quantities are in Arithmetical Proportion, the sum of the two extremes is equal to the sum of the two means. Thus, in the arithmeticals 4, 6, 7, 9, the sum 4 + 9=6+7 = 13: and in the arithmeticals a, a+d, b, b+d, the sum a+b+ d = a + b + d.

2. In any continued arithmetical progression, the sum of the two extremes is equal to the sum of any two terms at an equal distance from them.

Thus,