Hence, the number of shot in the pile will be equal to the sum of the series 1, 3, 6, 10, 15, 21, 28, 36; in which any term is found by adding 1 to the number of the term and multiplying the sum by half the number of terms. Thus, if we suppose the horizontal layers to be continued down, and denote the number of any layer from the top by n, we shall have and the sum of this series will express the number of balls in a triangular pile, of which n denotes the number in either of the bottom rows. If the general term of any increasing series of numbers involves n to the mth degree, the sum of the series will not involve n to' a higher degree than (m + 1). For, the sum of such series cannot exceed n times the general term, and hence, cannot involve n to a higher degree than m + 1. Let us therefore assume n (n + 1) 1 + 3 + 6 + 10 + 15 2 · = A + Bn + Cn2 + Dn3, in which the co-efficients A, B, C, and D, are not functions of n. In order that these co-efficients may be determined, we must find four independent equations involving them. If we make n = we have A+B+C+D=1 n = 2, gives A+ 2B + 4C8D=1+3 A3B9C + 27D=1+3+6 = 1 (1), = 4 (2), =10 (3), A4B16C + 64D1+3+6+10=20 (4). Now, by a series of subtractions we have Equation (2)-(1), gives B+ 3C + 7D 3... (5), third course by 32, &c. Hence, the series G will express the number of balls in a square pile, of which the number of courses, and consequently the number of balls in one of the lower rows is, n. To find the sum of this series, assume 1+ 4+ 9 + 16 + n2 = A + Bn + Cn2 + Dn3, = 14; A+ 4B16C + 64D1+4+9+16= and from these four equations, we find, by continued subtractions, D=, C, B, and A = 0; hence, Let us now suppose that we have a rectangular or oblong pile of shot, as represented in the figure below. Suppose we take off from the oblong pile the square pile EFD. We then see that the oblong pile may be formed by adding to the square pile a series of triangular strata, each containing as many balls as are contained in one of the faces of the square pile; and the number of the triangular strata will be one less than the number of balls in the top row. Therefore, if n denote the number of horizontal courses, the number of balls in one triangular strata n (n + 1); and if m+1 denotes the whole will be expressed by 2 number of balls in the top row, the number of triangular strata will be denoted by m; and the number of balls in all these strata X m. 17 But since the number of balls in a square pile, whose side contains n balls is the number of balls in an oblong pile, whose top row contains m+1 balls, and depth n balls, will be expressed by If we denote the general sum by S, we shall have the follow ing formulas for the number of shot in each pile. face of each pile, and the other factor, the number of balls in the longest line of the base plus the number in the side of the base opposite, plus the parallel top row, we have the following RULE. Add to the number of balls in the longest line of the base, the number in the parallel side opposite, and also the number in the top parallel row; then multiply this sum by one third the number in triangular face. EXAMPLES. 1. How many balls in a triangular pile of 15 courses? Ans. 680. 2. How many balls in a square pile of 14 courses? and how many will remain after 5 courses are removed? Ans. 1015 and 960. 3. In an oblong pile the length and breadth at bottom are respectively 60 and 30: how many balls does it contain ? Ans. 23405. 4. In an incomplete rectangular pile, the length and breadth at bottom are respectively 46 and 20, and the length and breadth at top 35 and 9: how many balls does it contain ? Ans. 7190. Summation of infinite Series. 247. An infinite series is a succession of terms unlimited in number, and derived from each other according to some fixed and known law. The summation of a series consists in finding an expression of a finite value, equivalent to the sum of all its terms. Different series are governed by different laws, and the methods of finding the sum of the terms which are applicable to one class. will not apply universally. A great variety of useful series may be summed by the following formula : If now, by attributing known values to p and q, and differen! values in succession to n, the expression n (n+p) shall repre sent a given series; then, the sum of this series will be equal multiplied by the difference between the two new series 1 P and the value of by ference of the sums of these series be known, p be known, we can find the value of the series (-s) even if we do not know the value of the new series and q EXAMPLES. 1. Required the sum of the series 1 1 1 + &c., to infinity. |