Theorem 1. In an infinite series of nom hers, increasing by an equal difference or ratio (that is, an arithmetical or geome trical increasing progression) from a gi ven number, a term may he found greater than any assignable number.
Hence, if the series increase by differ ences that continually increase, or by ra tios that continually increase, comparing each term to the preceding, it is mani fest that the same thing must he true, as if the differences or ratios continued equal.
Theorem 2. In a series decreasing in infinitum, in a given ratio, we can find a term less than any assignable fraction.
Hence, if the terms decrease, so as the ratios of each term to the preceding do also continually decrease, then the same thing is also true, as when they continue equal.
Theorem 3. The sum of an infinite se ries of numbers, all equal, or increasing continually, by whatever differences or ratios, is infinitely great ; that is, such a series has no determinate sum, but grows so as.to exceed any assignable number.
Demons. First, if the terms are all equal, as A : A : A, &c. then the sum of any finite number of them is the product of A by that number, as A n; but the greater n is, the greater is A n ; and we can take a greater than any assignable number, therefore A n will be still great er than any assignable number.
Secondly, suppose the series increases continually, (whether it do so infinitely or limitedly,) then its sum must be infinitely great, because it would be so if the terms continued all equal, and therefore will be more so, since they increase. But if we suppose the series increases infinitely, either by equal ratios or differences, or by increasing differences or ratios of each term to the preceding; then the reason of the sums being infinite will appear from the first theorem ; for, in such a se ries, a term can be found greater than any assignable number, and much more : therefore the sum of that and all the pre ceding.
Theorem 4. The sum of an infinite se ries of numbers decreasing in the same ratio is a finite number, equal to the quote arising from the division of the pro duct of the ratio and first term, by the ratio less by unity; that is, the sum of an assignable number of terms of the se ries can never be equal to that quote ; and yet no number less than it is equal is the value of the series, or to what we can actually determine in it; so that we can carry the series so far, that the sum shall want of this quote less than any assign able difference.
Demons. To whatever assigned number of terms the series is carried, it is so far finite ; and if the greatest term is 1, the least A, and the ratio r, then the sum is r / — A S See PROGRESSION.
Now, in a decreasing series from 1, the more terms we actually raise, the last of them A, becomes the lesser, and the less. er A is r 1 — A is the greater, and so al.
so is r 1- A• but r 1- A being still less r - 1 than r 4 therefore r /- A is still less than is, the sum of an assignable number of terms of the series is still less r / than the quote mentioned, which is and this is the first part of the theorem.
Again, the series may be actually con tinued so far, that r I - - A A shall want of r less than any assignable difference ; for, as the series goes on, A becomes less and less in a certain ratio, and so the series may be actually continued till A becomes less than any assignable number, (by The re om 2,) now — r r/ - 1- rl- A = A r - 1 r and r A is less than A ; therefore let any number assigned be called N, we can carry the series so far till the last term A be less than N; and because r A r - 1 wants of the r is less than A, which is also less than N, therefore the second part of the theorem is also true, and - is the true value of r - 1 the series.
Scholium. The sense in which r I / is called the sum of the series has been sufficiently explained ; to which, how ever, we shall add this, that whatever con sequences follow from the supposition of r / r being the true and adequate value of the series taken in all its infinite ca pacity, as if the whole were actually de termined and added together, can never be the occasion of any assignable error in any operation or demonstration where it is used in that sense : because, if it is said that it exceeds that adequate value, yet it is demonstrated that this excess must be less than any assignable differ. ence, which is in effect no difference, and so the consequent error will be in effect no error : for if any error can hap r I pen from being greater than it ought to be, to represent the complete value of the infinite series, that error de pends upon the excess / 1 over that complete value ; but this excess being unassignable, that consequent error must be so too ; because still the less the ex cess is, the less will the error be that de pends upon it. And for this reason we r I 1 may justly enough look upon ras ex pressing the adequate value of the infinite series. But we are further satisfied of the reasonableness of this, by finding, in fact, that a finite quantity does actually con vert into an infinite series, which happens in the case of infinite decimals. For ex ample, 3 = . 6 6 6 6, &e. which is plainly a geometrical series from 6 - in the con 6 6 tinual ratio of 10 to 1; for it is + 10 1U0 6 6 84c.