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This inexhaustibility, therefore, in the nature of numbers, is all that we can distinctly comprehend by their infinity; for though we can easily conceive that a finite quantity may become greater and greater without end, yet we are not, by that means, enabled to form any notion of the ultimatum, or last magnitude, which is incapable of farther augmentation.

Hence, we cannot apply to an infinite series the common notion of a sum, or of a collection of several particular numbers, which are joined and added together, one after another; as this supposes that each of the numbers composing that sum, is known and determined. But as every series generally observes some regular law, and continually approaches towards a term, or limit, we can easily conceive it to be a whole of its own kind, and that it must have a certain real value, whether that value be determinable or not.

Thus in many series, a number is assignable, beyond which no number of its terms can ever reach, or, indeed, be ever perfectly equal to it; but yet may approach towards it in such a manner, as to differ from it by less than any quantity that can be named. So that we may justly call this the value or sum of the series; not as being a number found by the common method of addition, but such a limitation of the value of the series, taken in all its infinite capacity, that, if it were possible to add all the terms together, one after another, the sum would be equal to that number.

In other series, on the contrary, the aggregate, or value of the several terms, taken collectively, has no limitation; which state of it may be expressed by saying, that the sum of the series is infinitely great; or, that it has no determinate or assignable value, but may be carried on to such a length, that its sum shall exceed any given number whatever.

Thus, as an illustration of the first of these cases, it

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may be observed, that if r be the ratio, g the greatest term, and the least, of any decreasing geometric series, the sum, according to the common rule, will be (rg-1)-(r-1) and if we suppose the less extreme, 1, to be diminished till it becomes 0, the sum of the whole series will be rg÷(r-1): for it is demonstrable, that the sum of no assignable number of terms of the series can ever be equal to that quotient; and yet no number less than it will ever be equal to the value of the series.

Whatever consequences, therefore, follow from the supposition of rg-(r—1) being the true and adequate value of the series, taken in all its infinite capacity, as if all the parts were actually determined, and added together, no assignable error can possibly arise from them, in any operation or demonstration where the sum is used in that sense; because, if it should be said that the series exceeds that value, it can be proved, that this excess must be less than any assignable difference; which is, in effect, no difference at all; whence the supposed error cannot exist, and consequently rg÷(r-1) may be looked upon as expressing the true value of the series, continued to infinity.

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We are, also, farther satisfied of the reasonableness of this doctrine, by finding, in fact, that a finite quantity is frequently convertible into an infinite series, as appears in the case of circulating decimals. Thus two-thirds expressed decimally is 3.66666 &c. 10 + 100 + 1000 +TOOOO + &c. continued ad infinitum. But this is a geometric series, the first term of which is, and the ratio; and therefore the sum of all its terms, continued to infinity, will evidently be equal to, or the number from which it was originally derived. And the same may be shewn of many other series, and of all circulating decimals in general.

With respect to the processes by which the summa

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tion of various kinds of infinite series are usually obtained, one of the principal is by the method of differences, pointed out and illustrated in prob. IV. next following.

Another method is that first employed by James and John Bernoulli, which consists in resolving the given series into several others of which the summation is known; or by subtracting from an assumed series, when put=s, the same series, deprived of some of its first terms; in which case a new series will arise, the sum of which will be known.

A third method, which is that of Demoivre, consists in putting the sum of the series =s, and multiplying each side of the equation by some binomial or trinomial expression, which involves the powers of the unknown quantity x, and certain known co-efficients; then, taking x, after the process is performed, of such a value that the assumed binomial, &c. shall become =0, and transposing some of the first terms, a series will arise, the sum of which will be known, as before.

Each of which methods, modified so as to render it more commodious in practice, together with several other artifices for the same purpose, will be found sufficiently elucidated in the miscellaneous questions succeeding the following problems.

PROBLEM 1.

Any series being given to find its several orders of dif ferences.

RULE.

1. Take the first term from the second, the second from the third, the third from the fourth, &c. and the remainders will form a new series, called the first order of differences.

2. Take the first term of this last series from the se

cond, the second from the third, the third from the fourth, &c. and the remainders will form another new series, called the second order of differences.

3. Proceed, in the same manner, for the third, fourth, fifth, &c. orders of differences; and so on till they terminate, or are carried as far as may be thought necessary. (u)

EXAMPLES.

1. Required the several orders of differences of the series 1, 2, 32, 42, 52, 62, &c.

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2. Required the several orders of differences of the series 1, 23, 33, 43, 53, 63, &c.

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3. Required the several orders of differences of the series 1, 3, 6, 10, 15, 21, &c.

Ans. 1st, 2, 3, 4, 5, &c. ; 2d, 1, 1, 1, &c. 4. Required the several orders of differences of the series 1, 6, 20, 50, 105, 196, &c.

Ans. 1st, 5, 12, 30, 45, 91, &c. ; 2d, 9, 16, 25, 36, &c.; 3d, 7, 9, 11, &c.; 4th, 2, 2, &c.

(u) When the several terms of the series continually increase, the differences will be all positive; but when they decrease, the differences will be negative and positive alternately.

5. Required the several orders of differences of the 1 1 1 1 1 2'4' 8' 16' 32'

series

&c.

PROBLEM II.

Any series, a, b, c, d, e, &c. being given, to find the first term of the nth order of differences.

RULE.

Let stand for the first term of the nth differences.

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n-2n-3

e &c. to n+1 terms, when n is an even

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n

-3

3

-e &c. to n + 1 terms, when n is an odd num

ber. (x)

EXAMPLES.

1. Required the first term of the third order of differerences of the series 1, 5, 15, 35, 70, &c.

Here a, b, c, d, e, &c. =1, 5, 15, 35, 70, &c. and n=3.

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(x) When the terms of the several orders of differences happen to be very great, it will be more convenient to take the logarithms of the quantities concerned, whose differences will be smaller; and, when the operation is finished, the quantity answering to the last logarithm may be easily found.

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