Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/247

From Wikisource
Jump to navigation Jump to search
This page needs to be proofread.


EXAMPLES

Show that the following ten series are convergent:

1. 6.
2. 7.
3. 8.
4. 9.
5. 10.

Show that the following four series are divergent :

11. 13.
12. 14.
 

142. Power series. A series of ascending integral powers of a variable, say , of the form

(A)

where the coefficients, are independent of , is called a power series in x. Such series are of prime importance in the further study of the Calculus.

In special cases a power series in may converge for all values of , but in general it will converge for some values of and be divergent for other values of . We shall examine (A) only for the case when the coefficients are such that

where is a definite number. In (A)

Referring to tests I, II, III, in §141, we have in this case , and hence the series (A) is

I. Absolutely convergent when , or ;
II. Divergent when , or ;
III. No test when , or .