十 + atd, a+2d, at3d &c. (1-2) 2 a+(a+d)x+(at2d)xa +(a+3d)x3 +(a+4d)x4 &c. And 2 = (1 – x)x{a+(a+d).+(+2d)x3 +(a +3d) x3&c.}=(1-x)a+dx, as will appear by actually multiplying by (1-2) Therefore z =(1-2a + dis; and consequently m atd. atd &c. sum of the infiinra (1-1) nite series required. a atat mr EXAMPLES FOR PRACTICE. 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 +32 +42 +52 &c. Ans. 42925 3. It is required to find the sum of the series 1+3x+ 6x3 + 10x3 + 15x4 continued ad infinitum, &c. when x is less than 1. 1 Ans. (1-2)3 4. It is required to find the sum of the series 1+4x2 10x2 +-20x3 +35x4 &c. continued ad infinitum, when x is less than 1. 1 Ans. (1-2)* 5. It is required to find the sum of the infipite series 1 1 1 5 1 + + st &c. Aps. 1.3 3:5 5.7 7.9 10' 2 6. Required the sum of 40 terms of the series (1 X 2); +(3X 4)+(6x6)+(7X8) &c. Aps, 22960 or + 2х + 2.3.4.5+ or 12' 2r-1 7. Required the sum of n terms of the series 2s 2x - 3, 2x - 5, 2x - 7 2. 24 Ans. n -&c. 1 8. Required the sum of the infinite series + 1.2.3.4 1 &c. Ans. 3.4.5.6 4.5.6.7 18 1 1 1 1 t-titat 4 10 20 1 3 1 &c. continued ad infinitum. Ans. 36' 10. It is required to find the sum of n terms of the series 1 +8+27.02 +6423 +125x4 &c. 1+4+39 Ans. (1-3)* ) 2 2 11. Required the sum of n term of the series -+ r2 3 4 5 6 1 1 nr+r-1 + + &c. 24 (n-1)2 (n-1)? 1 1 1 12. Required the sum of the series 十一十 2.6 4.8T 6.10 1 1 &c. (6) 8.12 4 ) 3 5n+-3n2 Ans. = s= 32 +48nt 16n? 1 1 1 13. Required the sum of the series, + 3.8 6.12 9,16 1 1 tes &c. 12.20 3n(4+4n) 1 Ans. S 12 +12n + på to&c. Ans. { rn + + 12 (6) The symbol 2, made use of in these, and some of the folJowing series, denotes the sum of an infinite namber of terms, and s the sum of n terms. + SS 6 6 6 14. Required the sum of the series 2.7+7.12+12.17 6 6 &c. + 17.22 (6n-3).(5n+2) 3 3n Aps. = 5' 2+on 1 1 1 15. Required the sum of the series 3.6 6.8 9.10 1 1 + &c. ... 12.12 3n(4+2n)" 1 Ans. >= 24' 2(3+6n 4(6+6n 2 3 4 16. Required the sum of the series 3.5 7.9 5 1+n + &c. 9.11 (1+2n).(3+2n) 1 1 12' 12 4(3+4n) 5 6 7 17. Required the sum of the series + ta 1.2.3' 2.3.4' 3.4.5 8 4+n ot &c. 4.5.6 =((n 1 2' 2-19n+2+n() . + (c) 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 Demoivre, Sterling's Method. Differ., James Bernoulli de Seri. Infin., Simpson's Math. Dissert., Waring's Medit. Analyt, Clark's translation of Lorgna's Series, the various works of Euter, and Lacroix Traité du Calgul Diff. et OF LOGARITHMS. (m) 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 and division of the natural numbers with which they are made to correspond. (d) 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 Int., where they will find nearly all the materials that have been hitherto collected respecting this branch of analysis. (d) 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, as had been done by Napier, but also by the publication, in 1624 and 1633, of his two great works, the Arithmetica Logarithmica, and the Trigonometria Britanica, both of which were formed upon the principle above mentioned ; as are, likewise, all our common logarithmic tables, at present in use. See, for farther details on this part of the subject, the Introduce tion 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 obtainedo raised, in order to produce all the common, or natural numbers. Thus, if ax=y, ax=y, a.*'=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 numbers. It is, likewise, farther evident, from the first of these equations, that when y=1, x will be =0, 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 wbich it belongs, is 1. Also, because aty, and ax'=y, it follows, from the multiplication of powers, that at x at, or attx'=yy' ; and consequently, by the definition of logarithms, given above, xtx'= log. yy, or log. yy=log. y +log. y'. And, for a like reason, if any number of the equations ax=y, a*=y', ax"=y", &c. be multiplied together, we shall have etti*'**"&c. yy'y" &c. ; and consequently *+x'+x"&c=log. yy'y' &c. ; or log. yy'y' &c. =log. yt. log. y' + log. y' &c. From which it is evident, that the logarithm of the product of any number of factors is equal to the sum of the logarithms of those factors, |