Radius and interval of convergence of power series

Welcome to the World of Power Series!

In your journey through Unit 10, you have seen many different types of series. Now, we are moving into one of the most powerful concepts in calculus: the Power Series. Think of a power series as a polynomial that never ends. Because these series contain a variable \(x\), they act like functions. But there is a catch—they don't always "work" (converge) for every value of \(x\). Our goal in this chapter is to find the "Safe Zone" where these series behave properly. This safe zone is defined by the Radius and the Interval of Convergence.

What is a Power Series?

A power series is an infinite series of the form:
\( \sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + \dots \)
In this expression, \(c\) is a constant called the center of the series, and \(a_n\) represents the coefficients.

Analogy: Imagine you have a flashlight (the series) centered at a specific spot \(c\). The light shines out to a certain distance. Inside that light, you can see clearly (the series converges). Outside that light, it is pitch black (the series diverges). Our job is to find how far that light reaches!

The Three Possibilities for Convergence

Every power series follows one of three rules regarding where it converges:

1. At a single point: The series converges only at its center \(x = c\). In this case, the Radius \(R = 0\).
2. Everywhere: The series converges for all real numbers \(x\). In this case, the Radius \(R = \infty\).
3. On a specific interval: The series converges if \(x\) is within a certain distance \(R\) from the center \(c\). This distance is the Radius of Convergence.

Key Takeaway: Every power series converges at its center. The question is simply how much further it goes.

How to Find the Radius of Convergence

To find the radius, we almost always use the Ratio Test (which you learned in Topic 10.8). Don't worry if you find the Ratio Test a bit intimidating; here is the step-by-step process:

Step 1: Set up the Ratio Test
Calculate the limit: \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n} \right| \).
Simplify the expression. You will usually be left with something like \( |x-c| \cdot (\text{some limit}) \).

Step 2: Set the limit less than 1
The Ratio Test tells us a series converges when the limit \(L < 1\). Set your simplified expression \( < 1 \) and solve for \( |x-c| \).

Step 3: Identify the Radius \(R\)
Once you have the inequality in the form \( |x-c| < R \), the number \(R\) is your Radius of Convergence.

Example: If you solve the Ratio Test and get \( |x-3| < 5 \), your center is \(3\) and your Radius \(R\) is \(5\).

From Radius to Interval of Convergence

The Interval of Convergence is the set of all \(x\)-values for which the series converges. Finding the interval is a two-part process:

Part A: Find the open interval
Using our example \( |x-3| < 5 \), we can write this as \( -5 < x-3 < 5 \). Adding \(3\) to all sides gives us \( -2 < x < 8 \). These are our "boundaries."

Part B: The "Endpoint Check" (Crucial Step!)
The Ratio Test is inconclusive when the limit equals exactly 1. This happens at the endpoints (in our example, \(x = -2\) and \(x = 8\)). You must plug these values back into the original series one at a time to see if they converge or diverge.

• Plug in the left endpoint and use your convergence tests (like the Alternating Series Test or p-Series Test).
• Plug in the right endpoint and do the same.
• If the series converges at an endpoint, use a bracket \([ \text{ or } ]\). If it diverges, use a parenthesis \(( \text{ or } )\).

Did you know? The radius is easy to find, but the interval is where students often lose points on the AP Exam because they forget to test the endpoints!

Summary Table for Intervals

If your center is \(c\) and your radius is \(R\), your interval will look like one of these four types:
• \( (c-R, c+R) \)
• \( [c-R, c+R) \)
• \( (c-R, c+R] \)
• \( [c-R, c+R] \)

Common Mistakes to Avoid

1. Forgetting Absolute Values: In the Ratio Test, the absolute value signs are mandatory. They ensure that even if the terms alternate in sign, we are looking at the distance from the center.

2. Mixing up Radius and Interval: The Radius is a distance (a single number like \(R=2\)). The Interval is a set of values (like \( -1 < x < 3 \)).

3. Misinterpreting \(R = \infty\): If the limit in your Ratio Test is always \(0\) (which is always less than \(1\)), then your Radius is infinity and your Interval is \( (-\infty, \infty) \). You don't need to check endpoints for infinity!

Quick Review

• Center (\(c\)): Where the power series is "anchored."
• Radius (\(R\)): The distance from the center to the edge of the convergence zone.
• Interval (\(I\)): The full list of \(x\)-values where the series works, including any "friendly" endpoints.
• Primary Tool: The Ratio Test is your best friend for finding \(R\).
• Secondary Tool: Use other tests (p-series, comparison, alternating) to check the endpoints individually.

Note: For more on specific tests like the Ratio Test or p-Series, see Topics 10.5 and 10.8. For more on creating these series, see Topic 10.14.