Unit 2: Power SeriesStudent ID:

Day 27 –Interval of Convergence

Opener:

Write the Maclaurin series for / Find the radius of convergence for using the Ratio Test
Will give a good approximation of ? Why or why not? / Will give a good approximation of ? Why or why not? Maybe use your calculator here!

Lesson:The Ratio Test is fairly straight forward for finding the Radius of Convergence. However, it is often useful to look at the full interval of values where your Taylor series is valid. Let’s start where you left off on part (b) of the opener:

Using Algebra, you should be able to isolate . However, having met you before, Mr. Grant fully understands that you may not remember how to solve an absolute value inequality. Remember that absolute value measures a number’s ______from . So, means “every number which is a solution for must be strictly less than 1 unit away from 0”. This is the interval of numbers from _____ to _____, or more commonly written as ____ _____. Therefore, any value of on that interval is valid to use in the Taylor seriesinstead of . Another way to state this is that any of those values will cause the Taylor Series to ______to the function . This idea is known as the ______of ______of a Power Series.

There is, however, one small issue with intervals of convergence: the ______of the interval. For example, at one end of the interval of , there is an asymptote and so it makes sense we wouldn’t include . However, at the other end of the interval is , and why shouldn’t that be included; it’s part of the function, continuous, differentiable, but yet we said that it shouldn’t be included. To be 100% sure about each endpoint, you are REQUIRED to substitute each endpoint into the original Power Series to justify why, or why not, you discarded the endpoints in your answer.

Endpoint for

For , ______because: / Endpoint for

For , ______because:

Find the radius and interval of convergence for each Power Series. Don’t forget to check the endpoints!

Time for some Multiple Choice practice. Please show your work for each problem, using another piece of paper if necessary. No calculator should be used unless otherwise specified.

What are all values of for which the following series converges?

The radius of convergence for the power series below is equal to . What is the interval of convergence?



The radius of convergence of the series is
/ / / / Infinite
What are all the values of for which the following series converges?

What is the interval of convergence of the power series below?

What is the radius of convergence for the power series below?

The power series below has a radius of convergence .

At which of the following values of can the alternating series test be used with this series to verify convergence at ?
What is the radius of convergence for the series

The power series below converges conditionally at . Which of the following statements about convergence of the series at is true?

The series converges absolutely at .
The series converges conditionally at .
The series diverges at .
There is not enough information given to determine convergence of the series at .