One meaning of sum of a periodic series con tinued ad infinitain," would evidently be the product of the value of the period, and an infinite integer, or the period added to itself ad It may, however, be considered, that if continued ad inf. every term of the period has as strong a claim as its last term to be considered as the last term of the series. The sum of a periodic series, whose period = 0, continued ad i2 f. is therefore susceptible of as many different values as there are different terms in its period. If the last term of the series be the first term of the period, that term will be equal to the sum. If the last term of the series be the second term of the period, the sum of the series will be the sum of the first two terms of the period; and if the last term of the series be the third term of the period, the sum of the series will be the sum of the first three terms of the period, and so on. It is, therefore, evident, that the variety of values of which the sum of the series is susceptible, whether continued ad inf. or not, is limited by the number of different terms in the period.
Now the sum of the series continued ad inf. is sometimes said to be a mean of all its different values, or to be the sum of all the different values divided by their number. We shall give an instance of this in the series of which we have already obtained the sum.
If n' be the number of different terms in the series, we shall obtain the n' different values of the sum by successively substituting 0, 1, 2, - - - - n' for n in the equation.
cos (A 4 x) cos (A + 2 n ± 1 2 x) = 2 sin 2 x. S Let S' be the sum of all the corresponding values of S. Since the sum of all the corresponding values of cos (A + 2 n + is the period, which by hyp. = 0 we have n' cos (A x) = 2 sin 2 x . S' S' cos (A z x) 2 sin .4 x This is the value of S which would be obtained by neglecting the last cosine in the investigation already instituted, and it hence appears that we cannot infer that the sum of the series ad i2 f. has this value, ex cept when the series is periodic, and its period = 0, and even then the sum of the series has this value only in the sense above explained, which is, that the sum ad inf. is susceptible of as many different values as there are terms in the period, and that which is found above is its mean value, or the sum of all its different values divided by their number.
In general, however, the summation of a series con tinued ad inf. is, properly speaking, an analytical pro cess wich is the reverse of development. As develop ment is the process by which a function expressed in finite terms is converted into a series, so summation is that process by which, when a series is given, the function by whose development it was obtained may be assigned. The word summation cannot be literally applied to this process, except in the case where the series converges, for, except in this case, there cannot be an arithmetical equality between the function and the series. This will very evidently appear by an ex ample. Let I be divided by (Iu) by the ordinary rules of algebraical division, and we obtain = 1 + a + a 3 + at- - - - all inf.
If a be greater than unity, the second member of this equation is an infinitely great positive quantity, while the first member is a finite negative quantity. In this case no :Actual equality can subsist, and the sign only signifies that the first member is the function by whose development the series in the second member is obtained.
The method used in the last examples may some times be applied to the summation of a finite number of terms of a numerical series as in the following example.
Let the series be 1 1 2 2 n 1 2 /1 = (2 71 1) (2 71+ 1) The sum of the first members of these equalities is evidently 1 2 since all the other terms mutu 1 ally destroy each other, and the sum of the second members is 2 S. Hence we obtain 2 n 2 S = 1 2n+1 2n+1 . S 2a+ I But by far the most general method for the summa tion of series is that which is derived from the prin ciples of the calculus of differences, which we shall now briefly explain and illustrate by examples.
If the successive terms of a series be expressed by Ste. the number at the foot of the letter denoting the place of the term, the sum of all the terms from u, to inclusive, may be expressed thus, S fix. Thus we have S = 2z, + -1- . . +u,.
In like manner S + +n, +. uz . Subtracting the former from the latter, we have S uz+2 -f- uz+ 3 + n