And reversely, if we take this series, and find its sum by the preceding theo 6 rem, it comes to the same 4 ; for / = 60 r = 10, therefore r I = 6; and r r / 6 2 - 1 = 9; whence - - r - 1 - 9 - 5' We have added here a table of all the varieties of determined problems of in finite, decreasing geometrical progres sions, which all depend upon these three things, viz. the greatest term 4 the ratio r, and the sum S ; by any two of which the remaining one may be found : to which we have added some other pro blems, wherein S - L is considered as a thing distinct by itself, that is, without considering S and L separately.
Theorem 5. In the arithmetic progres sion 1, 2, 3, 4, &c. the sum is to the pro duct of the last term, by the number of terms, that is, to the square of the last term, in a ratio always greater than 1 : 2, but approaching infinitely near it. But if the arithmetical series begins with 0, thus, 0, 1, 2, 3, 4, &c. then the sum is to the product of the last term, by the number of terms, exactly in every step as 1 to 2.
Theorem 6. Take the natural progres sion, beginning with 0, thus, 0, 1, 2, 3, &c. and take the squares of any the like powers of the former series, as the squares, 0, 1, 4, 9, &c. or cubes, 0, 1, 8, 27: and then again take the sum of the series of powers to any number of terms, and also multiply the last of the terms summed by the number of terms, (reck oning always 0 for the first term,) the ra tio of that sum, to that product, is more than n X 1' (ii being the index of the powers,) that is, in the series of squares it is more than 3; in the cubes more than ; and so on : but the series going on in infinitum, we may take in more and more terms, without end, into the sum ; and the more we take, the ratio of the sum to the product mentioned grows less and less ; yet so as it never can actually 1 1 be equal to n X but approaches infi nitely near to it, or within less than any assignable difference.
"The nature, origin, &c. of series." Infinite series commonly arise, either from a continued division, or the extrac tion of roots, as first performed by Sir I. Newton, who also explained other ge neral ways for the expanding of quanti ties into infinite series, as by the binomi al theorem. Thus, to divide 1 by 3, or to expand the fraction 3 into an infinite series; by division in decimals in the or dinary way, the series is 0.3333, &c. or
3 3 3 3 &c. where 1 + 100 ± -r 10000' the law of continuation is manifest. Or, if the same fraction 3 be set in this form 1 2 and division be performed in the algebraic manner, the quotient will be 1 1 1 1, 1 , 1 1 , 3 = = 1- a' &c. Or, if it he expressed in this form, 1 1 by a like division there will 3 4 1 arise the series, 1 1 1 1 1 1 1 "ifi &c. =." + &c. And thus, by dividing 1 by 5-2, or 6 3, or 7 4, &c. the series answering to the fraction 4. may be found in an end less variety of infinite series ; and the fi nite quantity 4 is called the value or ra dix of the series, or also its sum, being the number or sum to which the series would amount, or the limit to which it would tend or approximate, by slimming up its terms, or by collecting them toge ther one after another. In like manner, by dividing 1 by the algebraic sum a + c, or by a c, the quotient will be in these two cases as below, viz.
11 c , c' c3 acc.
a + c a a' a3 1 1 c3 &e.
a c c a a3 where the terms of each series are the same, and they differ only in this, that the signs are alternately positive and nega tive in the former, but all positive in the latter.
And hence, by expounding a and c by any numbers whatever, we obtain an end less variety of infinite series, whose sums or values are known. So by taking a or c equal to 1, or 2, or 3, or 4, &c. we ob tain these series, and their values; 1 1 1 + 1 7 1 1 + 1 1+1.-1, &c.
I 1 , ± - = 3-1 2 3 3' ±9 1 1 1 1 1 1 2 -rn-1 1 2 + 2' 23, &c. 1 1 1 1 1 1 = 3 S3 34' And hence it appears, that the same quantity or radix may be expressed by a great variety of infinite series, or that many different series may have the same radix, or sum.
Another way in which an infinite series arises, is by the extraction of roots. Thus, by extracting the square root of the number 3 in the common way, we ob tain its value in a series as follows, viz.. %/ 3 = 1.73205, &c.
= 1 + 7 3 10 100 2 s 1000 100000' &c. in which way of resolution the law of the progression of the series is not visible, as it is when found by division. And the square root of the algebraic quantity a' ? c' gives = a Ta &c.