Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/249

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Substituting in (B), we get

which is an alternating series that converges.

Substituting in (B), we get

which is convergent by comparison with the p series ().

The series in the above example is said to have as the interval of convergence. This may be written , or indicated graphically as follows:


Wag-142-1 Interval of Convergence


EXAMPLES

For what values of the variable are the following series convergent?

Graphical representation of intervals of convergence[1]
15. Ans. Wag-142-2 Interval of Convergence
16. Ans. Wag-142-3 Interval of Convergence
17. Ans. Wag-142-4 Interval of Convergence
18. Ans. Wag-142-5 Interval of Convergence
19. Ans. All values of . Wag-142-6 Interval of Convergence
20. Ans. All values of . Wag-142-6 Interval of Convergence
21. Ans. All values of . Wag-142-6 Interval of Convergence
22. Ans. All values of . Wag-142-6 Interval of Convergence
  1. End points that are not included in the interval of convergence have circles drawn around them.