Join Our Discord

Learn Moreย ย ๐Ÿ‘‡

caret down
caret up
discord-logo

Join our Discord

1,000s of Fiveable Community students are already finding study help, meeting new friends, and sharing tons of opportunities among other students around the world! Don't miss out!ย  ๐ŸŽ‰

group of students
group of students

โ™พ๏ธย ap calc

ย ย >ย ย 

โ™พย Unit 10

ย ย โ€ขย ย โฑ๏ธ1 min read

10.13 Radius and Interval of Convergence of Power Series

Peter Cao

peter cao

โฑ๏ธ June 8, 2020

๐Ÿ“…โŒจ๏ธ

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. 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

A tent in the woods with a fire

Get your ap calc survival pack!

Download our ap calc survival pack and get access to every resource you need to get a 5.

ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

continue learning

Slide 1 of 17
Fiveable

Join Our Discord

discord-logo

Fiveable Community students are already meeting new friends, starting study groups, and sharing tons of opportunities for other high schoolers. Soon the Fiveable Community will be on a totally new platform where you can share, save, and organize your learning links and lead study groups among other students!๐ŸŽ‰

*apยฎ and advanced placementยฎ are registered trademarks of the college board, which was not involved in the production of, and does not endorse, this product.

ยฉ fiveable 2021 | all rights reserved.

2550 north lake drive
suite 2
milwaukee, wi 53211