Q264, power series expansion of a rational function

TL;DR

Find the power series expansion for the function 2x^5 / (9 + x^2).

Transcript

it didn't have the endpoint okay we're going to find a power series expansion for function 2x to the fifth power over nine plus x squared and as we can see this is just a rational function right so of course we can utilize our best friend and remember for our best read we must have a one on the top that this is the 2x to the fifth power so let's lo... Read More

Key Insights

• ✊ The power series expansion is a useful tool for approximating functions.
• 👋 The best friend method helps ensure that the power series is correctly formatted.
• ✊ The radius of convergence determines the domain of validity for the power series.

Questions & Answers

Q: How is the power series expansion obtained for the given rational function?

The power series expansion is obtained by factoring out a term from the denominator and manipulating the expression to fit the form required by the best friend method.

Q: What is the significance of the best friend method in finding the power series?

The best friend method ensures that the power series is in the correct form, with the input raised to the nth power and a positive coefficient.

Q: How is the radius of convergence determined?

The radius of convergence is determined by setting the absolute value of the input (in this case, x^2) to be less than 1 and solving for x.

Q: Why are the endpoints of the interval of convergence not included?

When using algebraic manipulation to find the interval of convergence, the endpoints are not included in the final interval. This is a general rule when using the best friend method.

Summary & Key Takeaways

• A power series expansion is derived for the function 2x^5 / (9 + x^2) using algebraic manipulation.

• The best friend method is used to ensure the power series is in the correct form.

• The radius of convergence is determined to be 3, and the interval of convergence is (-3, 3).