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Unit 10.13 – Radius and Interval of Convergence of Power Series BC ONLY

AP® Calculus BC | Where Does the Series Work? Understanding the Power of Series

Why This Matters: Knowing **where** a power series converges is just as important as knowing what it equals! The *radius* and *interval* of convergence tell you the “domain” of a power series, defining which \(x\)-values give meaningful results. This is the foundation for all applications of power series in calculus and physics.

🎯 Power Series Overview

Definition of Power Series

General Form:
\[ \sum_{n=0}^{\infty} c_n (x-a)^n \]

Centered at \(a\), with coefficients \(c_n\).

📝 Tip: If \(a=0\), it's a Maclaurin series.

🔎 What Is the Radius of Convergence?

The Radius of Convergence

Definition:
\[ \sum_{n=0}^{\infty} c_n (x-a)^n \] converges for \( |x-a| < R \), diverges for \( |x-a| > R \)

The value \(R\) is called the **radius of convergence**.

  • **Inside** (\(|x-a| < R\)): Converges absolutely
  • **Outside** (\(|x-a| > R\)): Diverges
  • **At Endpoints** (\(|x-a| = R\)): Must check individually!

📏 How to Find the Radius & Interval

Main Methods

1. Ratio Test (most common):
\[ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \]
Converges if \(L < 1\)
2. Root Test:
\[ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \]
Converges if \(L < 1\)
Process:
  1. Write \(a_n\) as function of \(x\) (usually \(c_n(x-a)^n\))
  2. Apply ratio test or root test
  3. Solve for values of \(x\) that make \(L < 1\)
  4. The solution gives the **interval**; the distance to the center is the **radius**
  5. ALWAYS CHECK BOTH ENDPOINTS SEPARATELY!

📚 Summary Table: Finding Convergence

Convergence Tests for Power Series
Test When to Use General Steps
Ratio Test Almost all power series Simplify terms, limit \(\left|\frac{a_{n+1}}{a_n}\right|\), solve for \(x\)
Root Test When terms involve \(n^{\text{th}}\) powers Limit \(\sqrt[n]{|a_n|}\), solve for \(x\)
p-Series/Alternating/Comparison Test Checking endpoints only! Plug endpoint into series, apply test

📖 Worked Examples

Example 1: Ratio Test with Maclaurin Series

Series: \(\sum_{n=1}^{\infty} \frac{x^n}{n}\)

Apply Ratio Test:

\[ L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left|\frac{x^{n+1}/(n+1)}{x^n/n}\right| = \lim_{n \to \infty} \left| x \cdot \frac{n}{n+1} \right| = |x| \]

Converges if \(|x| < 1\), diverges if \(|x| > 1\), check endpts!

Conclusion: Radius \(R = 1\), interval: \(-1 < x < 1\)

Example 2: General Series Centered at a ≠ 0

Series: \(\sum_{n=0}^\infty \frac{(x-2)^n}{3^n}\)

\[ L = \lim_{n\to\infty} \left| \frac{(x-2)^{n+1}/3^{n+1}}{(x-2)^n/3^n} \right| = \left| \frac{x-2}{3} \right| \]

Converges if \( |x-2| < 3 \), diverges if \( |x-2| > 3 \)

Radius: \(R = 3\), Interval: \( -1 < x < 5 \) (check endpoints!)

Example 3: Endpoints Matter

Series: \(\sum_{n=1}^{\infty} \frac{(x+2)^n}{n \cdot 4^n}\)

\[ a_n = \frac{(x+2)^n}{n\cdot 4^n} \]
\[ L = \lim_{n\to\infty} \left| \frac{(x+2)^{n+1}/((n+1)4^{n+1})}{(x+2)^n/(n4^n)} \right| = \lim_{n\to\infty} \left| \frac{x+2}{4} \cdot \frac{n}{n+1} \right| = \left| \frac{x+2}{4} \right| \]

So radius \(R=4\), interval \(-6 < x < 2\).
At \(x=-6\): plug in, get \(\sum \frac{(-4)^n}{n\cdot 4^n} = \sum \frac{(-1)^n}{n}\), which converges.
At \(x = 2\): get \(\sum \frac{4^n}{n\cdot 4^n} = \sum \frac{1}{n}\), which diverges.

Final interval: \([-6,2)\)

💡 Fast Steps and Tricks

  • Always use absolute value: Ratio/root tests use \( |x-a| \)!
  • After finding R: State interval as \(a-R < x < a+R\)
  • Plug endpoints into ORIGINAL series: Use p-series/comparison/alternating/convergence tests to check!
  • For geometric series: \(\sum r^n\) has radius \(|r|<1\) ⇒ \(R=1\)
  • If limit L has no x: Converges everywhere or nowhere!
  • If L is always zero: Series converges for all x, so \( R = \infty \)
  • Always check both endpoints separately!
  1. Write out next term \(a_{n+1}\).
  2. Form ratio \(\frac{a_{n+1}}{a_n}\), simplify completely.
  3. Take limit as \( n \to \infty \), reduce to \( |x-a| < R \).
  4. State radius and interval clearly.
  5. Plug in both endpoints to check separately!

❌ Common Mistakes to Avoid

  • Forgetting to use absolute value in the ratio/root test.
  • Not simplifying the ratio BEFORE taking limit.
  • Assuming endpoints always converge (must check!).
  • Thinking radius is always 1 (it's not!)
  • Ignoring interval shift for series centered at \(a \neq 0\).
  • Forgetting to test both endpoints

📝 Practice Problems

Find radius and interval of convergence for:

  1. \(\sum_{n=1}^{\infty} \frac{(2x)^n}{n}\)
  2. \(\sum_{n=1}^{\infty} \frac{(x-3)^n}{5^n}\)
  3. \(\sum_{n=1}^{\infty} \frac{x^n}{n!}\)
  4. \(\sum_{n=1}^{\infty} \frac{(x+1)^n}{n2^n}\)
  5. \(\sum_{n=1}^{\infty} \frac{(x-2)^{2n}}{n4^n}\)

Answers:

  1. Radius 0.5, interval \(-0.5 < x < 0.5\), check endpoints
  2. Radius 5, interval \(-2 < x < 8\), check endpoints
  3. Radius \(\infty\), interval \((-\infty, \infty)\)
  4. Radius 2, interval \(-3 < x < 1\), check endpoints
  5. Radius 2, interval \(0 < x < 4\), check endpoints (\(x-2\) squared!)

⚡ Quick Reference Formulas

  • Power series: \(\sum_{n=0}^{\infty} c_n (x-a)^n\)
  • Converges for: \( |x-a| < R \)
  • Find R with ratio test:
    \[ R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right| \]
  • Interval of convergence: \( (a-R, a+R) \); test endpoints!

Master power series convergence! The radius of convergence tells you how far from the center you can go and have the series work; the interval of convergence tells you exactly which values of \(x\) the series converges for. Always check endpoints separately. Use the ratio or root test for R; state interval as \(a-R < x < a+R\). Common tricks: geometric-like? R=1. Factor blessed? Likely R=∞. Most frequent error: forgetting to check endpoints! Appears on every BC exam—be systematic, bold the radius, and show endpoint checks!