๐ Topic Summary
The interval of convergence for a power series determines the set of $x$ values for which the series converges. It's found using the ratio or root test, and then checking the endpoints of the resulting interval for convergence (conditional or absolute) or divergence. This gives you the complete interval where the series converges to a finite value.
๐๏ธ Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Radius of Convergence |
A. The set of all $x$ values for which a power series converges. |
| 2. Interval of Convergence |
B. A series where the terms alternate in sign. |
| 3. Power Series |
C. A series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$. |
| 4. Alternating Series |
D. Half the length of the interval of convergence. |
| 5. Absolute Convergence |
E. Convergence of $\sum |a_n|$. |
(Answers: 1-D, 2-A, 3-C, 4-B, 5-E)
โ๏ธ Part B: Fill in the Blanks
To find the interval of convergence, we often use the ________ Test or the ________ Test. After finding the interval, it is crucial to check the ________ because the test is inconclusive there. Convergence at an endpoint may be ________ or ________.
(Answers: Ratio, Root, Endpoints, Conditional, Absolute)
๐ค Part C: Critical Thinking
Explain why checking the endpoints of an interval obtained using the ratio test is a necessary step when determining the interval of convergence. Provide an example to illustrate your explanation.
