For example, the sum of the first x terms of the series
14+-¢+4+0°4-0°4 ..- is 1—2
If x is numerically less than 1, the sum approaches to the finite limit; 1 1-x and the series is therefore convergent.
If x is numerically greater than 1, the sum of the first n terms is x^n-1 x-1, and by taking n sufficiently great, this can be made greater than any finite quantity; thus the series is divergent.
If x=1, the sum of the first n terms is n, and therefore the series is divergent.
If x=-1, the series becomes
1—1+4+1-—141-1+4-..
The sum of an even number of terms is zero, while the sum of an odd number of terms is 1; and thus the sum oscillates between the values 0 and 1. This series belongs to a class which may be called oscillating or periodic convergent series.
469. When the Sum of the First n Terms of a Given Series is Unknown. There are many cases in which we have no method of finding the sum of the first » terms of a series. We proceed therefore to investigate rules by which we can test the convergency or divergency of a given series without effecting its summation.
470. First Test. An infinite series in which the terms are alternately positive and negative is convergent if each term is numerically less than the preceding term.
Let the series be denoted by
1b) = Us +~- Ug — Uy -F Uh, — Uy ==
where le) > thy > Ay Ul ln
