153. Now let m=1, then (a+b)=(a+b)2 = √ a+b;

and a =a; hence the series in Art. 152 is transformed into

= 1 + 2 + 1 = 7 + ( 1 − r ) ( 1 − 2r) + (1 − r) (1 − 2r) (1 − 3 r)

√2=1+-+

2.373

√2:

2

3*

2.3.4.7

7

24 27 28

3+

By means of the series marked (A), the rth root of many other numbers may be found, if a and b be so assumed, that b is a small number with respect to a, and Va a whole number; thus,

EXAMPLE 1.

Let a=4, b=1, r=2, then Va=√4=2, and we have

Let a=8, b=1, r=3, then Va=V8=2, and we obtain

The several terms of these series are found by substituting for a, b, and r their values in the general series marked (A) or (B), and then rejecting the factors common to both the numerators and denominators of the fractions. Thus, for instance, to find the seventh term of the series exhibiting the value of √2, we take the 7th term of the series marked (B), which is (1 − r) (1 — 2 r)(1 − 3 r) (1-4) (1—5"); and since r=2, the

2.3.4.5.67.6

r)

3.7 2.4.2.2"

3.7 To find the 5th term of the scries express

ing the approximate value of 9, we take the 5th term of the

general series marked (A), which is

(-)

(1-7) (1-2r) (1-3r)

2.3.47*

where a=8, b=1, and r=3; .. the value of the

2.5 In this manner each term of the several series is cal

35.8+

culated; and the law which they observe is, that the numerators of the fractions consist of certain combinations of prime numbers, and the denominators of combinations of certain powers of a and r.

154. These series converge very fast, so that a few terms would give the rth root of certain numbers with a great degree of accuracy. But a more practical method of finding the higher roots of such numbers, is, by making the number whose root is

to

to be extracted equal to a+b, and then assuming a+x=

a+b, x being some decimal fraction; for in this case (a+x)* =a+b, and by expanding (a+x) and neglecting all the powers of x after x' (being very small compared with the preceding ones) we have

a2+ra' ̄'x+r('"'———1) a ̃¬x"=a+b;

•'.ra^'x+r('— 1) a'~'x'=b (A) an equation

2

from which the value of x may be found in two ways.

I. By arranging the terms, and dividing by r('—–—1)a”~,

which is HALLEY'S Rule, (Philosophical Transactions, 1694).

By a first approximation, neglecting the term which involves x, we

b rar

and we obtain a second approximation which

the Rule given by LA CROIX (Complémens d'Algébre), and ascribed to LAMBERT.

EXAM. 1.

Find an approximate value of the cube root of 67.

Now 67-64+3=4'+3; .. a=4, b=3, r=3; hence,

Find an approximate value of the fifth: root of 30.

Here 30=32-2=23 — 2; .'. a=2, b = −2, r=5; hence,

--

The method of finding the rth root of certain numbers as exhibited in this and the foregoing Article, is a matter rather of curiosity than practical utility, as the rth root of any number what, ever may be found with great facility by means of Logarithms. This method would be useful, however, in an operation where it was required to express this root in the form of a vulgar fraction; as in the last Example, where we obtained the approximate value of the 5th root of 30 in the shape of the fraction

310

157

XLIV.

On the method of finding the approximate Ratio of the Powers and Roots of Numbers whose Difference is small.

155. Let a +x and a be two numbers whose difference is x,

n(n-1)

n(n-1)(n-2)

2

2.3

then (a+x)" a" :: a"+na"-'x+ a"-3x+&c. a" :: (dividing each term of the ratio by a)

156. Suppose now that n is not a large number, and that x is very small when compared with a, then the fractions.

a+nx:a.

2
x2 x3

&c.

" 29 a a

will be small also, and those terms in which they are involved will be very small when compared with the integral part anx of the series; in this case, therefore, the ratio of (a + x)" : a" approximates to the ratio of a+nr: a. Thus the ratio of (a+x)': a' approximates to the ratio of a+2x: a; of (a+x)3 a to the ratio of a+ 3x a ; &c. &c.; or if n=1, 3, &c. then the ratio of √a+x: a approximates to the ratio of a+x: a; of a +x: a to the ratio of a+x: a; &c. &c. For instance, the ratio of the square of 501 to the square of 500 (in which case, a=500, x=1, n=2) is 502: 500 very nearly; the ratio of the cube of 62 to the cube of 61, is 64: 61 very nearly; &c. &c. Again, the ratio of the square root of 501 to the square root of 500 is 500: 500; and of the cube root of 103 to the cube root of 100, is 101: 100, very nearly.

157. If the difference between the two numbers is not very small when compared with the numbers themselves, then the three first terms of the series must be taken instead of two, in which case the approximate ratio of (a+x)" : a" becomes that of n(n: a. For instance, let it be required to

a+nx +

2