13

and the convergency begins from the 23rd term: if n =

and

4

4

we have a point of convergency when r = 3, and, sub

3

sequently, a point of divergency, when r = 17: if n = − 12 and

10 we find r = 5, or the divergency begins with the 6th

9

term of the series.

for (1 + x)" 1, and (1 − 1)′′

are

are conver

gent or

693. It appears, therefore, that the series for (1 + x)" and The series (1-x)" are or become, convergent, whenever a is less than or whenever the binomials, whose powers are developed, arithmetical, both in their arrangement and value: but that divergent according they cease to be so whenever a is greater than 1, or whenever as 1+r and they cease to be arithmetical either in their value or their are arrangement: for in this case 1 -x is negative, and therefore cal or not, unarithmetical in its value: whilst under the same circumstances, their arthe binomial 1+a, though arithmetical in its value, is not so in rangement its arrangement, inasmuch as the greater term succeeds the less in an inverse and interminable operation (Art. 386)*: but it

and is greater than 1, we may obtain positive values of r from both these formula, which shews that a point of convergency is in such a case followed by a point of divergency.

It is in the inverse operations of Arithmetic and Arithmetical Algebra, such as Division and Evolution, that we meet with interminable results, and in which the arrangement of the terms of the expressions, which are the subjects of the operations, in the order of their magnitude, is absolutely necessary to enable us to approximate to, when we cannot accurately obtain, the true result which is sought for: such an arrangement, which is emphatically called arithmetical, is not absolutely necessary, though generally convenient, in operations, whether direct or inverse, which lead to a terminable result.

Thus

arithmeti

both in

and value.

should be observed, that in the latter case, the terms of the resulting series, either are, or become, alternately positive and negative, confirming the remark which has elsewhere been made (Arts. 517 and 650), that the expression in which it originates, though arithmetical in its value, is not so in the character of the operation to which it is subjected.

694. If the series which arises from the developement of (1 + x)" be arithmetical, or convergent, we shall be enabled not only to approximate to its true value or sum (s) by the aggregation of its terms (provided we include the term from which the convergency begins), but likewise to assign the limits of the excess or defect of the aggregate thus formed from the true sum which is required.

Thus if σ be the aggregate or sum of r terms of the series, and T, with its proper sign, its (1 + r)th term, then the true sum s will differ from σ by a quantity less than

Thus (1+r)2 = 1 + 2x + x2 and (x + 1)2 = x2 + 2x + 1, and the results

1+2x+2 and +2x+1

are identical in their arithmetical value, though not in their arrangement, whether a be less or greater than 1: but (1 + x)−2 = 1 − 2x + 3x2 - 4x3 + ... and

and if x be less than 1, it is the first, and if r be greater than 1, it is the second, of these series which is convergent: or, in other words, the convergency or divergency of the resulting series depends upon the arithmetical or unarithmetical arrangement of the terms of the binomial.

For if = 1.

n+1

P

it will follow, since 1

n+1

increases as r increases, that the sum of the geometrical series T-Tpx + Tp2x2 - Tp3 3 + ...,) which is

T

(Art. 432), will be less than the sum o' of the corresponding terms 1 + px

of the series for (1 + x)" ; but the sum of the geometrical series

T-Tx + Tr2- Tx3 + ...,

will be greater than o' since p is necessarily less than 1, when r is

greater than n it follows, therefore, inasmuch as 'is intermediate in value between

As an Example, suppose it was required to determine within Example. what limits of error the square root of 5 would be given by

the aggregation of 5 terms of the series for 2(1+1), or

and therefore s differs from 2.2360687 by a quantity less than

CHAPTER XXIII.

The signs

of affection hitherto required

and recog

nized, are

the four biqua

ON THE USE OF THE SQUARE AND HIGHER ROOTS OF 1 AS

are +, -,

SIGNS OF AFFECTION.

695. THE signs of affection, which we have hitherto used, √-1 and -1; the two first of which+ and may be conveniently replaced by +1 and -1, if we consider them, like the two last, as factors of the symbols which they affect*. Thus + a + 1 x a and a = ~ - 1 x a, in the same manner

dratic roots that a√-1=√1 × a and

of 1.

Their use in gene

operations

metical Algebra.

-a

- 1

=√1xa (Art. 652).

Of these signs, the two first + 1 and − 1, or 1 and − 1, are the

two square roots of 1: whilst the two last, -1 and

are the two square roots of 1: the entire series of them, + 1, -1, +√√-1 and -/-1, may be easily shewn to be the four biquadratic roots of 1†.

696. The preceding signs, or their equivalents, are sufralizing the ficient, as we have shewn in the preceding Chapters, to express ordinary every symbolical consequence which arises from extending the of Arith- rules for Addition, Subtraction, Multiplication, Division, and the Extraction of the square root, which are proved in Arithmetical Algebra, to all values whatsoever of the symbols to which they are applied, in conformity with the general principle of the permanence of equivalent forms (Art. 631): no further signs of affection would be required, if the processes of Arithmetical and Symbolical Algebra, were confined to the several operations above enumerated.

Corresponding use of the higher orders of

697. But the processes of Evolution in Arithmetic and Arithmetical Algebra, are not confined to the extraction of square and biquadratic roots, and it will be found necessary to introthe roots of duce new signs, in order to give the same extension to the rules for the extraction of cubic and higher roots, and which, like those we have already considered, are capable, as we shall proEvolution. ceed to shew, of being correctly expressed or symbolized by the multiple symbolical values of the cubic and higher roots of 1.

⚫ 1 in the

higher pro

cesses of

See Appendix.

† For r1- 1 = (x2o − 1) (x2 + 1 ) = 0: the roots of the equation - 1 = 0, are 1 and -1: those of x2 + 1 = 0, are √1 and

=

698. Thus a3=1 × a3, and therefore Ja3/1a: inasmuch as we have shewn (Art. 669), that the three roots of 1 are

it will follow that there are also three cube roots of a3, which are

roots of - 1.

699. In a similar manner, we have - a3-- 1 × a3, and there- Of the cube fore-a-1 × a: and inasmuch as we have shewn that the three cube roots of 1 (Art. 670), are

it will follow that there are also three cube roots of a3, which

-

roots of 1

700. It may be easily shewn that the three cube roots of The cube 1, and the three cube roots of -1, form the six senary roots and -- 1 of 1: for if x = 1, we have x = 1, and therefore

are the senary roots of 1.