If in the series [2) we make (1=0, a' = I and (1.1, wc obtain, I 3 (*) 10 Hence we find I 6 102 + s= ' " - = 4 2 ' 1 3 610 57 χ Ti + ) - 1 + 4.. + - - - = a and so on.
This method of summation by addition was propos ed by James Bernoulli, in a tract De Scriebus published with his .Irs Conjectandi. Bas. 1713.
Another method of summation suggested by the Bernoullis on the same principle, is summation by subtraction, of which the following is an example.
To ,find the sum of an infinite series of reciprocal tri angular numbers. Let the series be In the preceding examples, we have applied these methods of addition and subtraction only to cases in which the series were continued in infinitum. We can, however, frequently employ them for the summa tion of a finite number of terms of a series. Nume rous instances of this occur in trigonometry. The following are examples of it.
Ex. To find the sum of the sines of a series of arcs in arithmetical progression.
Let the proposed series be Sin A + sin (A ± x) sin (A 4. 2 x) - - Sin [A + (n 1) al = S. [1] Let both members be multiplied by 2 sin 1 x.
by substituting successively 0, 1, 2, S, - - - 1), for in in this formula we shall obtain the values of the successive terms of the above series, and it is plain that except the first and last they will mutually de stroy each other; so that the result will be It will be perceived that the artifice by which the summation of this series has been brought tinder the methods of addition and subtraction, is by converting it into another series, every term of which being dou ble the product of two sines, admits of being resolved into two simple cosines, with different signs. In this case, one of the cosines into which each term is re solved, destroys one of the cosines into which the next term is resolved; so that however numerous the terms series may be, the total result can only contain one of the cosines of the first pair, and one of the last pair. If the last cosine continually diminished or approached any value as a limit, as the number of terms in the series increases, we should be entitled to conclude that the sum of the proposed series continu ed ad infinitum, would be expressed by the first cosine and the limiting value of the last. we assume that
the sum of the series is expressed by the first term alone, it is equivalent to assuming that the last term diminishes without limit.
This, however, is not the case; the last cosine al ternately increases and diminishes, and changes its sign as the arc changes its relation to an exact mul tiple of the circumference; and therefore the series increases and decreases alternately, and approaches no limiting state. In other words, the series not be ing convergent, does not admit of having its sum as signed when the number of its terms are limited.
If a. be commensurable with the circumference, or 2 Fr, the series will be periodic; that is, after a certain number of terms, the same terms will continually re cur in the same order. Let the least integers in the ratio of x to 2 rtbe m' : 11.'; so that 2 2/2',7.
In that case, when the series [1] has been continued to n' terms, the (n' )th, term will he it'± Sin (A + 2 nex, = sin (A + 2 m'r )= sin A. In like manner, the following term will be Sin [A + (n'+ 1) x] = sin ( A +x + 2 sin ( A +x,) which is equal to the second term of the series, and so the terms from the (ie+ 1)th to the 2 Oh inclusive, will be equal to those from the first to the n'th inclu sive.
In this case, the value of the period of the series 2 nen may be found by substituting n' for x, and n' for n in the value of S, already found; which gives 1) .
Sin (A ) sill 1n'7 n' S Sin If be not an integer, the value of S must = 0, for n' rn' sin = 0, and sin cannot = O. Therefore, in this case, the terms of the period mutually destroy each other, or the whole period might be divided into two periods, the terms of which differ only in their In' signs. But if 711. be an integer, then sin = 0; 7/ and the value of S assumes the form In this case x is an exact multiple of the circumference, and the terms of the series arc all equal to sin A, which is it self the period.