EXAMPLES. 1. Required the sum of n terms of the series succession, will produce the given series. The two corresponding auxiliary series, to n terms, are The difference between the sums of n terms of the first and of n terms of the given series by S, we have, we 2. Required the sum of n terms of the series If we compare the terms of this series with the expression see that p = 2, q = 1, and n = 1, 3, 5, 7, &c., in suc cession. 3. Required the sum of n terms of the series Here p = 3, q=1, n = 1, 2, 3, 4, &c. The two auxiliary series, to n terms, are, 4. Required the sum of the series 4 4 5. Find the sum of n terms of the series, 4 + &c. Ans. 1. If n is even, the upper sign is used, and the quantity in the last parenthesis becomes + 1, in which case If n is odd, the lower sign is used, and the quantity in the last parenthesis becomes 0, in which case 6. Find the sum of n terms of the series, Of the Method by Differences. 209. Let a, b, c, d &c., represent the successive terms of a series formed according to any fixed law; then if each term be subtracted from the succeeding one, the several re mainders will form a new series called the first order of dif ferences. If we subtract each term of this series from the succeeding one, we shall form another series called the second order of differences, and so on, as exhibited in the annexed If, now, we designate the first terms of the first, second, third, &c. orders of differences, by da, da, da, da, &c., we shall de-4d6c-4b+a, whence e = a + 4d, + 6d1⁄2 + 4dz + dâ, &c. &c. &c. &c. And if we designate the term of the series which has n terms before it, by T, we shall find, by a continuation of This formula enables us to find the (n+1)th term of a series when we know the first terms of the successive orders of differences. 210. To find an expression for the sum of n terms of the series a, b, c, &c., let us take the series 0, a, a + b, a+b+c, a+b+c+d, &c. (2). (3). Now, it is obvious that the sum of n terms of the series (3), is equal to the (n+1)th term of the series (2). But the first term of the first order of differences in series (2) is a; the first term of the second order of differences is the same as d in equation (1). The first term of the third order of differences is equal to do, and so on. Hence, making these changes in formula (1), and denoting the sum of n terms by S, we have, When all of the terms of any order of differences become equal, the terms of all succeeding orders of differences are 0, and formulas (1) and (4) give exact results. When there are no orders of differences, whose terms become equal, then formulas do not give exact results, but approximations more or less exact according to the number of terms used. EXAMPLES. 1. Find the sum of n terms of the series 1.2, 4.5, &c. 2.3, 3.4, Hence, we have, a 2, d, 4, d1 = 2, da, da, &c., equal = = = |