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377 Rh 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