Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/244



Our results may then be stated as follows: Given the series of positive terms

find the limit

This is the name given to a series whose terms are alternately positive and negative. Such series occur frequently in practice and are of considerable importance.

If

is an alternating series whose terms never increase in numerical value, and if

then the series is convergent. Proof. The sum of $$2n$$ (an even number) terms may be written in the two forms

Since each difference is positive (if it is not zero, and the assumption $$\lim_{n \to \infty}u_n = 0$$ excludes equality of the terms of the series), series (A) shows that $$S_{2n}$$ is positive and increases with $$n$$, while series (B) shows that $$S_{2n}$$ is always less than $$u_1$$; therefore, by Theorem I, §136, $$S_{2n}$$ must approach a limit less than $$u_1$$ when $$n$$ increases, and the series is convergent.

Test the alternating series $$1 - \tfrac{1}{2} + \tfrac{1}{3} - \tfrac{1}{4} + \cdots$$

Solution. Since each term is less in numerical value than the preceding one, and

the series is convergent.

A series is said to be absolutely or unconditionally convergent when the series formed from it by making all its terms positive is convergent. Other convergent series are said