678 SERIES term -( 3 + 5 + 6), where for the same values of n the series will be 1, 2, 4, 8, 15, 26, . . The series may contain nega- tive terms, and in forming the sum each term is of course to be taken with the proper sign. 2. But we may have a given law, such as either of those just mentioned, and the question then arises, to find the sum of an indefinite number of terms, or say of n terms (71 standing for any positive integer number at pleasure) of the series. The expression for the sum cannot in this case be obtained by actual addition ; the formation by addition of the sum of two terms, of three terms, <fec., will, it may be, suggest (but it cannot do more than suggest) the expression for the sum of n terms of the series. For instance, for the series of odd numbers 1+3 + 5 + 7 + ..., we have 1 = 1, 1+3 = 4, 1 + 3 + 5 = 9, &c. These results at once suggest the law, 1 + 3 + 5 . . . + (2 - 1) = 2 , which is in fact the true expression for the sum of terms of the series ; and this general expression, once obtained, can afterwards be verified. 3. We have here the theory of finite series : the general problem is, u n being a given function of the positive integer n, to determine as a function of n the sum u o + v i + u z + M or i* 1 or der to have n instead of n + 1 terms, say the sum u + u } +-u 2 . . + _ 1. Simple cases are the three which follow, (i.) The arithmetic series, a + (a + b) + (a + 2b) . . + (a + n - 1 )b ; writing here the terms in the reverse order, it at once appears that twice the sum is = 2a + n - Ib taken n times : that is, the sum = na + ^n(n 1 )6. In particular we have an expression for the sum of the natural numbers 1 + 2 + 3. .. + n = ^n(n + l), and an expression for the sum of the odd numbers 1 + 3 + 5. . + (2n-l) = u 2 . (ii.) The geometric series, a + ar+aj a ,..+ar n ~ l ; here the difference between the sum and ? times the sum is at once seen to be = a - ar n , and the sum is thus = r^ ; in particular the sum of the series (iii.) But the harmonic series, !+_L i a a + b a + 2b'" J or say + + "> does not admit of summation; there is no algebraical function of n which is equal to the sum of the series. 4. If the general term be a given function ii n , and we can find v n a function of n such that v n +i - Vn ^n, then we have u = v l - v , v l = v 2 - v v u 2 = v 3 - v^ . .u^ = v+i - v n ; and hence + : + u 2 . . + v, n = v n +i - V Q , an expression for the required sum. This is in fact an application of the Calculus of Finite Differences. In the notation of this calculus v n + - v n is written Av n ; and the general inverse problem, or problem of integration, is from the equation of differences Av n == u n (where u n is a given func- tion of n) to find v n . The general solution contains an arbitrary constant, v n = V n + C ; but this disappears in the difference v n+ i V Q . As an example consider the series ), = 871(71 + 1 ), here, observing that n(n + 1 )(n + 2) - (n - 1 )n(n + 1 ) = n(n + l)(n+H - n^ we have r re +i and hence 1 + 3 + 6 . . + in( + 1) = |n( + l)(n + 2), as may be at once verified for any particular value of n. Similarly, when the general term is a factorial of the order r, we have r + 1 71(71 + 1) ..(n + r-l)_n(n + l)..(n + r) 1 ' 1.2 .. r 1.2 .. (r + 1)' 5. If the general term u n be any rational and integral function of n, we have 7!. 71(71-!).., n(n-l) . . (n-p + 1) p = o + 1 A o + 2 AX - + - - i. 2 . . /, A 7 'o> where the series is continued only up to the term depend- ing on />, the degree of the function u n , for all the subse- quent terms vanish. The series is thus decomposed into a set of series which have each a factorial for the general term, and which can be summed by the last formula ; thus we obtain 1.2 (n + l)(-l)..(n-j + l ) ?> . 1.2.3.. (p+1) which is a function of the degree p + 1. Thus for the before-mentioned series 1 + 2 + 4 + 8+ .., if it be assumed that the general term u n is a cubic function of n, and writing down the given terms and forming the differences, 1, 2, 4, 8 ; 1, 2, 4 ; 1,2; 1, we have _ n n(ti-l) n(n-l)(n-2) f _1 1 1.2 1.2.3 and the sum u . , , " . , + ti n . (n + l)n( 1.2 1.2. 3 1.2.3.4 71 + 24). As particular cases we have expressions for the sums of the powers of the natural numbers .. + n 3 = (observe that this = (l + 2 . . . +) 2 ) ; and so on. 6. We may, from the expression for the sum of the geometric series, obtain by differentiation other results : -, l-r n thus l+7- + 7- 2 ...+7- n ~= - gives and we might in this way find the sum U Q + uj' . . . + u n r n , where u n is any rational and integral function of n. 7. The expression for the sum u + u l . . . +u n of an in- definite number of terms will in many cases lead to the sum of the infinite series U Q + T . . . ; but the theory of infinite series requires to be considered separately. Often in dealing apparently with an infinite series u + u-^ + ... we consider rather an indefinite than an infinite series, and are not in any wise really concerned with the sum of the series or the question of its convergency : thus the equation really means the series of identities j 7 n , 7t(>t-l) . Tl+ 1.2 ' 1.2 1.2 obtained by multiplying together the two series of the left-hand side. Again, in the method of generating func- tions we are concerned with an equation <(?) = A Q + Aj. . . . + A n t n + .., where the function 4>(t) is used only to ex- press the law of formation of the successive coefficients. It is an obvious remark that, although according to the original definition of a series the terms are considered as arranged in a determinate order, yet in a finite series
