πŸ“š

Β >Β 

♾️ 

Β >Β 

β™Ύ

10.13 Radius and Interval of Convergence of Power Series

1 min readβ€’june 7, 2020

Athena_Codes

Athena_Codes


AP Calculus AB/BC ♾️

279Β resources
See Units

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(565).png?alt=media&token=7742581e-e75a-447c-bc60-5bc5a9ef5dd8
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. Let’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
Browse Study Guides By Unit
πŸ‘‘Unit 1 – Limits & Continuity
πŸ€“Unit 2 – Fundamentals of Differentiation
πŸ€™πŸ½Unit 3 – Composite, Implicit, & Inverse Functions
πŸ‘€Unit 4 – Contextual Applications of Differentiation
✨Unit 5 – Analytical Applications of Differentiation
πŸ”₯Unit 6 – Integration & Accumulation of Change
πŸ’ŽUnit 7 – Differential Equations
🐢Unit 8 – Applications of Integration
πŸ¦–Unit 9 – Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)
β™ΎUnit 10 – Infinite Sequences & Series (BC Only)
πŸ“šStudy Tools
πŸ€”Exam Skills

Fiveable
Fiveable
Home
Stay Connected

Β© 2024 Fiveable Inc. All rights reserved.


Β© 2024 Fiveable Inc. All rights reserved.