CHAPTER XLI.
Convergency And Divergency Of Series.
466. We have, in Chapter xxxiv., defined a series as an expression in which the successive terms are formed by some regular law; if the series terminates at some assigned term, it is called a finite series; if the number of terms is unlimited, it is called an infinite series.
In the present chapter, we shall usually denote a series by an expression of the form
u_1 + u_2 + u_3 + ... + u_n + ...
467. Definitions. Suppose that we have a series consisting of n terms. The sum of the series will be a function of n; if n increases indefinitely, the sum either tends to become equal to a certain finite limit, or else it becomes infinitely great.
An infinite series is said to be convergent when the sum of the first n terms cannot numerically exceed some finite quantity, however great n may be.
An infinite series is said to be divergent when the sum of the first n terms can be made numerically greater than any finite quantity by taking n sufficiently great.
TESTS FOR CONVERGENCY.
468. When the Sum of the First n Terms of a Given Series is Known. If we can find the sum of the first n terms of a given series, we may ascertain whether it is convergent or divergent by examining whether the series remains finite, or becomes infinite, when n is made indefinitely great
