# Power Series and Interval of Convergence (BC Calculus - Unit 10)

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### Description

**Power Series and Interval of Convergence Lesson:**

Your **AP Calculus BC students** will use the definition of a power series, find the radius and interval of convergence, and determine the endpoint convergence of power series.

**What is included in this resource?**

⭐ Guided Student Notes

⭐ *Fully-editable *SMART Board® Slides

⭐ Homework/Practice assignment

⭐ Content quiz

⭐ Full solution set

**Lesson Objective:**

★ Understand the definition of a power series

★ Find the radius and interval of convergence of a power series

★ Determine the endpoint convergence of a power series

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**NOTE: This lesson has been updated to reflect the changes in the AP Calculus CED Binder .**

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**College Board® Topics:**

Topic 10.13: Radius and Interval of Convergence of Power Series

**College Board® Learning Objectives:**

**LIM-8.D:** Determine the radius of convergence and interval of convergence for a power series. **(BC Only)**

**LIM-8.D.1: **A power series is a series of the form ∑(an (x-r)^n)

where n is a non-negative integer {an} is a sequence of real numbers, and r is a real number. **(BC Only)**

**LIM-8.D.2:** If a power series converges, it either converges at a single point or has an interval of convergence. **(BC Only)**

**LIM-8.D.3:** The ratio test can be used to determine the radius of convergence of a power series. **(BC Only)**

**LIM-8.D.4:** The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence. **(BC Only)**

**LIM-8.D.5:** If a powe3r series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval. **(BC Only)**

**LIM-8.D.6:** The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series. **(BC Only)**

**The unit includes the following topics:**

2. The Integral Test and p-Series

6. Taylor Polynomials and Approximations

7. Lagrange Error Bound and Accuracy

8. Power Series and Interval of Convergence

9. Taylor and Maclaurin Series

10. Representing Functions by Power Series

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**AP® is a registered trademark of the College Board® which was not involved in the creation of, and does not endorse this product.**

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