CHAPTER XLIV.
Summation Of Series.
513. Examples of the summation of certain series (Arithmetic and Geometric) have occurred in previous chapters. We will now consider methods for summing other series.
514. Recurring Series. A series , in which from and after a certain term each term is equal to the sum of a fixed number of the preceding terms multiplied respectively by certain constants, is called a recurring series. A recurring series is of the 1st, 2d, or rth order, according as 1, 2, or r constants are required as multipliers.
515. Scale of Relation. In the series
,
each term after the second is equal to the sum of the two preceding terms multiplied respectively by the constants 2x and -x^3; these quantities being called constants because they are the same for all values of x. Thus
;
that is, ;
and generally, when n is greater than 1, each term is connected with the two that immediately precede it by the equation
,
or, .
In this equation the coefficients of u_{n}, u_{n-1}, and u_{n-2}, taken with their proper signs, form what is called the scale of relation.
