Examples. Find by the difference method the nth term and the sum of n terms of each of the following: 1. 2, 6, 12, 20, 30, 42, ..... Ans. T„=n(n+1); Sn= 3 (n+1) (n+2). 2. 4, 12, 28, 52, 84, 124, ..... Ans. Tn=4(n2−n+1); Sn= · 4n (n2+2). 3. 2, 9, 28, 65, 126, 217, ..... 4. 6, 24, 60, 120, 210, 336, n 3 Ans. T„=n3+1; Sn= 77 (n3+2n2+n+4). Ans. Tn=n(n+1)(n+2); 5. 4, 18, 48, 100, 180, 294, . 6. 2, 9, 20, 35, 54, .... 4 n 12 Ans. T„=n(n+1)2; Sn = 11⁄2 (n+1)(n+2) (3n+5). n Ans. Tn=(n+1)(2n−1); Sn= 1⁄2 (4n2+9n−1). 7. -1, 0, 15, 56, 135, 264, 455,..... n 6 Ans. Tn=n(2n−1)(n−2) ; Sn= (n+1)(3n2 — în+1). 6 8. Find the term midway between the first and second in Example 6 and Example 7. 9. Find the sum of the squares of the first n integers, and the number half-way between the first and second terms. 10. Given f(x) = x3 — 5x2+4x−3, show that the fourth order of differences of the series f(0), f(1), f(2), f(3), f(4), f(5) is zero. Find by interpolation f(2.9) and f(0.5). 11. Given the series 0.5176, 0.6840, 0.8452, 1.0000, 1.1472, 1.2856, the lengths of the chords subtending 30, 40, 50, 60, 70 and 80° at the center of a circle of radius 1.0000; difference the series, and by interpolation find to four decimals the chords corresponding to 35° and 46°. Ans. 0.6014 and 0.7814. 150. Sum of an Infinite Series-Convergence and Divergence. -In the case of the infinite geometric series, we have seen that if the common ratio r is less than unity in numerical value, the sum of n terms approaches a definite limit as n is indefinitely increased, or, in formulæ, 151. Definitions.-When Sn, the sum of n terms of a series, approaches a definite limit S as the number of terms, n, is indefinitely increased, the value S is called the sum of the series. The series is said to be convergent, and to converge toward the number S. Any series which is not thus convergent is called a divergent series. For example, a geometric progression in which r is numerically greater than unity is divergent because Sn=a доп -1 -1 can be made greater than any assignable quantity by making n large enough. 152. Convergence Unaffected by Omitted Terms. It is evident that if the first few terms of an infinite series be removed, the infinite series left will converge or diverge according as the original series converges or diverges, and vice versa. 153. Absolutely Convergent Series. A convergent series which contains only positive terms, or a convergent series which remains convergent after the signs of all its negative terms are changed, is called an absolutely convergent series, and is said to converge absolutely. 5 5 For instance, 10, -5, 5, -,,-, etc., is a geometric progression of which the common ratio, -, is >-1 and <1, and is therefore convergent. Moreover, since the series 10, 5, §, 4, §, .... is also convergent, the first series converges absolutely. 154. Alternating Series.-If the terms of an infinite series eventually alternate in sign, and become smaller and smaller the further the series progresses, the n-th term approaching zero as a limit as n is indefinitely increased, the series is convergent. This proposition is almost self-evident; the general proof is the same as for the following example: Consider the series 1, -,, −1, 1, −1, .. Plot points to represent the successive values of Sn: The odd S's move toward the left, the even toward the right; the difference between the inner two at any time is equal to the numerical value of the term last taken (i. e., == when n terms 1 n have been taken), and so approaches zero as n is indefinitely increased. The intermediate value upon which the S's are closing in is the sum, S, of the series. If the terms did not alternate at the start, the discussion would be the same, the sum of the terms before the alternating begins taking the place of 0 in the figure. A similar study of the series 2, −1, 1, −13, .... shows that, though its terms alternate in sign and decrease constantly, it is an oscillating divergent series, because its nth term does not approach zero as a limit as n=∞. The series 1, 1, 1, −1, 1, −, .... is not absolutely convergent, because the series 1, 1, 1, 1, .... diverges, as appears if we group its terms as follows: 1 + 1 > 1 ; +1 >2(1), or 1; į + b + t + >4(1), or 1; ++ .... ++>8(1%), or ; and so on. Out of the terms of this series, we can take as many groups of terms as we please, each group amounting to more than ; consequently, we can make Sn as large as we please by taking n large enough. 155. Region and Limits of Convergence. We have seen what is meant by the convergence or divergence of a numerical series; when the variable of a variable series is given a particular constant value, the resulting numerical series may, in most cases, be either convergent or divergent, according to the value of the constant so substituted. For instance, we know that the geometric series 3x+3x2+3x3+.... will furnish a convergent series if any positive or negative proper fraction be put in place of x, but a divergent series if any other number be substituted for x. The values of its variable for which a variable series becomes convergent form its region of convergence; the boundaries of this region are called its limits of convergence. A series may or may not converge for its limits of convergence; in the case of the geometric series, the limits of convergence are -1 and +1, for which values the series diverges. 156. Use of Convergent Series.-The sum of a variable infinite series has a definite value for every value of the variable within the region of convergence; and so, within that region, expresses a function of the variable. Outside of that region, the series has no sum and no meaning, and is useless. The sum of the first few terms of an infinite series is a polynomial; it can be shown that if the series is absolutely convergent, the results obtained by using this polynomial are approximately the same as if the function defined by the whole series had been used. The approximation can be made as close as desired by taking enough terms. In this way, when a function is expanded into a series, its value can be computed for any given value of its variable; expressions can also be formed for the sum, product, quotient, etc., of any functions expressed in series. As series are simpler and easier to handle than any other expression for some functions, and are the only means of expressing others, they are very important. Most of the practical work of mathematics is performed with the aid of series. A series that converges, but not absolutely, is of little use; its sum can be changed by merely rearranging its terms; moreover, such a series converges very slowly. Example. (by actual division) into a series in two ways, =1+x−x2+x3 −x1+ 1+2x = Show that the region of convergence of the first series is between 1 and +1, and the region of convergence of the second series is outside of this. Compare the values of S, of the first series with the values of when x, 0.1, 0.01, respectively, and the values of S4 of the second series with the 1+2x values of when x=2, 10, 100, respectively. 1+x 157. Ratio of Convergence.-The notation introduced in the study of the progressions is used for all series; namely, Tz for the kth term, T1, T2, T3, for the first, second, third, terms, .... .... The ratio of any term, T+1, to the next preceding, Tn, is called the ratio of convergence, or the test-ratio of the series; it |