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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:
EXAMPLES
For what values of the variable are the following series convergent?
Graphical representation of intervals of convergence[1] | |||
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15. | Ans. | ![]() | |
16. | Ans. | ![]() | |
17. | Ans. | ![]() | |
18. | Ans. | ![]() | |
19. | Ans. All values of . | ![]() | |
20. | Ans. All values of . | ![]() | |
21. | Ans. All values of . | ![]() | |
22. | Ans. All values of . | ![]() |
- ↑ End points that are not included in the interval of convergence have circles drawn around them.