SUMMATION OF SERIES 9. For summation of series, and for difference-equations gen erally, the advancing-difference notation is usually the more suitable. There are two useful methods.
(i.) Suppose we want to find the value of such a sum as . • • Let us denote this sum by Then we see that = — = 2 = O.
Thus the fourth and higher differences of the S's are all o. Now construct the table of differences (Table VI.).
operation is easily performed. The above is an example of a difference-equation. Another simple form is where a is a constant. This is equivalent to au,,, so that the solution is obviously Can, where C is an arbitrary constant. A more general form is the linear equation with constant coefficients un+p+aiun+P—i+a2u, ..2+ • • • = N, where ' • • • ap are constants, and N is a given function of n.