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may be observed, that if r be the ratio, g the greatest term, and l the least, of any decreasing geometric series, the sum, according to the common rule, will be (rg—l)-3-(r-1); and if we suppose the less extreme, l, 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 seTheS. - -
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 ry--(r-1) may be looked upon as expressing the true value of the series, continued to infinity.
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 #–.66666 &c.- or + rj + Torr + roorg + &c. continued ad infiniturn. But this is a geometric series, the first term of which is or, and the ratio 1's ; 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.
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 ar, and certain known co-efficients ; then, taking ar, 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.
Any series being given to find its several orders of dif. ferences.
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. k
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 neces
sary. (u) ExAMPLES. - A
1. Required the several erders of differences of the series 1, 2*, 32, 42, 52, 62, &c. 1, 4, 9, 16, 25, 36, &c. 3, 5, 7, 9, 11, &c. 1st diff. 2, 2, 2, 2, &c. 2d diff. - 0, 0, 0, &c. 3d diff.
2. Required the several orders of differences of the series 1, 2*, 3°, 48, 5*, 6°, &c. 1, 8, 27, 64, 125, 216, &c. 7, 19, 37, 61, 91, &c. 1st diff. 12, 18, 24, 30, &c. 2d diff. 6, 6, 6, &c. 3d diff. 0, 0, &c. 4th diff.
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.
(n) 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.
Any series, a, b, c, d, e, &c., being given, to find the first term of the nth order of differences.
Let 3 stand for the first term of the nth differences.
1. Required the first term of the third order of differ. erences 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.
(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.
+ 125=0; so that the first term of the fourth order is 0. 3. Required the first term of the eighth order of dif. ferences of the series, 1, 3, 9, 27, 81, &c. (y) Ans. 256
4. Required the first term of the fifth order of differ
To find the nth term of the series, d, b, c, d, e, &c. when the differences of any order become at last equal to each other.
Let d', d", d", div, &c. be the first of the several orders of differences, found as in the last problem.
(y) The labour, in questions of this kind may be often abridged, by putting ciphers for some of the terms at the beginning of the series ; by which means several of the differences will be equal to 0, and the answer, on that account, obtained in fewer terms.