Then, (1 x)2 And z = =x+2x2+3x+4x+5x, &c. (1 − x)2 × (x + 2x2 + 3x3 + 4x + 5x3, &c.) - Whose sum is =x+0+0+0+0,&c. Or + + + + &c., + = 2 sum 2 4 8 16 32 64' of the infinite series required. 16. It is required to find the sum (s) of the series + + + &c., continued ad infinitum. 27 81 (1 − x)3 × (x + 4x2 + 9x3 + 16xa, &c.) = x+x3, as will be found by actual multiplication. 17. Required the sum (s) of the series a + 2d a + 3d + mr2 &c., continued ad infinitum. α a+d m + mr mr3 a + (a + d) x + (a + 2d) x2 + (a + 3d) x3 + (a + 4d)x1, &c. And z = (1-x)3× {a + (a + d) x + (a + 2d) x2+(a+3d)x3, &c., } = (1 − x) a + dx, as will appear by actually multiplying by (1 - x)3. a Therefore, z= (1-x) a + dx; and consequently, + m = sum of the infinite series required. EXAMPLES. 1. Required the sum of 100 terms of the series 2, 5, 8, 11, *14, &c. Ans. 15050. 2. Required the sum of 50 terms of the series 1 + 22 + 3a +42+53, &c. Ans. 42925. 3. It is required to find the sum of the series 1 + 3x + 6x2+10x3+15x1, continued ad infinitum, &c., when x is less than 1. Ans. 1 (1-x)3. 4. It is required to find the sum of the series 1 + 4x + 10x+20x3+35x1, &c., continued ad infinitum, when x is less than 1. 1 1 5. It is required to find the sum of the infinite series, + + + &c. 1.3 3.5 5.7 7.9' 6. Required the sum of 40 terms of the series (1 × 2) +(3 × 4)+(5 x 6) + (7 x 8), &c. Ans. 86884. 2x - 1 7. Required the sum of n terms of the series 2x + 1 1 1 1 9. Required the sum of the series+ + + + 10 20 35' 3 1 Ans. or 1 2' 2' &c., continued ad infinitum.` 10. It is required to find the sum of n terms of the series 1+8x+27x2+64x3 + 125x1, &c., continued ad infinitum. 11. Required the sum of n terms of the series + 1 (1 x) 4 1 (r− 1)2 1 12. Required the sum of the series + + + *The symbol Σ, made use of in these and some of the following series, denotes the sum of an infinite number of terms, and S the sum of n terms. LOGARITHMS are a set of numbers that have been computed and formed into tables, for the purpose of facilitating many difficult arithmetical calculations; being so contrived, that the addition and subtraction of them answers to the multiplication *The series here treated of are such as are usually called algebraical, which, of course, embrace only a small part of the whole doctrine. Those, therefore, who may wish for farther information on this abstruse but highly curious subject, are referred to the Miscellanea Analytica of of Demoivre, Sterling's Method Differ., James Bernouilli's de Seri. Infin., Simpson's Math. Dissert., Waring's Medii Analyt., Clark's translation of Lorgna's Series, the various works of Euler, and Lacroix Traite du Calcul. Diff. ct Int., where they will find nearly all the materials that have been hitherto collected respecting this branch of analysis. and division of natural numbers with which they are made to correspond.* Or, when taken in a similar but more general sense, logarithms may be considered as the exponents of the powers to which a given or invariable number must be raised, in order to produce all the common, or natural numbers. Thus, if a2 = y, a21 = y', ax'' = y'', &c. then will the indices x, x', x", &c., of the several powers of a, be the logarithms of the numbers y, y', y', &c., in the scale, or system, of which a is the base. So that from either of these formulæ it appears, that the logarithm of any number, taken separately, is the index of that power of some other number, which, when involved in the usual way, is equal to the given number. And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural number. It is likewise farther evident, from the first of these equations, that when y 1, x will be = O, whatever may be the value of a; and consequently, the logarithm of 1 is always 0, in every system of logarithms. = And if x 1, it is manifest from the same equation, that the base a will be = y; which base is therefore the number whose proper logarithm, in the system to which it belongs, is 1. *This mode of computation, which is one of the happiest and most useful discoveries of modern times, is due to Lord Napier, Baron of Merchiston, in Scotland, who first published a table of these numbers in the year 1614, under the title of Canon Mirificum Logarithmorum; which performance was eagerly received by the learned throughout Europe, whose efforts were immediately directed to the improvement and extensions of the new calculus that had so unexpectedly presented itself. Mr. Henry Briggs, in particular, who was, at that time, professor of geometry in Gresham College, greatly contributed to the advancement of this doctrine, not only by the very advantageous alteration which he first introduced into the system of these numbers, by making 1 the logarithm of 10, instead of 2.3025852, has had been done by Napier; but also by the publication, in 1624 and 1633, of his two great works, the Arithmetica Logarithmica and the Trigonometrica Britanica, both of which were formed upon the principle abovementioned; as are, likewise, all our common logarithmic tables at present in use. See, for farther details on this part of the subject, the Introduction to my Treatise of Plane and Spherical Trigonometry, 8vo. 2d edit., 1813; and for the construction and use of the tables, consult those of Sherwin, Hutton, Taylor, Callet, and Borda, where every necessary information of this kind may be readily obtained. |