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Unit 10

10.13 Radius and Interval of Convergence of Power Series

1 min readโ€ขjune 8, 2020

peter-cao

Peter Cao


AP Calculus AB/BCย โ™พ๏ธ

Bookmarkedย 7.1kย โ€ขย 259ย resources
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One of the questions we have about power series approximations of functions is where the approximation is valid, or in other words, where the power series converges. For a given x, we can find the radius, and then the interval of convergence for a power series.
For a Taylor series centered at x = a, the only place where we are sure that it converges for now is x = a, but we can expand this to a greater range using our knowledge of the ratio test. Letโ€™s make an example to demonstrate this!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(800).png?alt=media&token=4b9635ca-a34c-49c0-b99e-b49b84b20a88
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(802).png?alt=media&token=dea3379b-d86a-4a1d-9743-19ac636a1216
We have to test the endpoints by plugging them into x as sometimes it may converge or diverge at the endpoints.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(805).png?alt=media&token=d217b0a9-165d-449e-b592-36d584981318
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(806).png?alt=media&token=7c4c809f-bda1-4c46-aa8b-0275d89a6964

Problems

Find the interval, radius of convergence, and the center of the interval of convergence.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(808).png?alt=media&token=70dbddb3-5474-495d-9de3-8cb4744da66d

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