AP Calculus: Power Series – Radius and Interval of Convergence

Power Series – Radius and Interval of Convergence

Concept Summary

• Power Series: An infinite sum of the form ? a?(x – c)?, representing a function as a polynomial with infinitely many terms; central to approximating transcendental functions.
• Radius of Convergence (R): The distance |x – c| < R within which the power series converges absolutely; R = 0 (converges only at x = c), R = ? (converges for all x), or 0 < R < ? (converges on an interval).
• Interval of Convergence (IoC): The set of x values where the power series converges, always symmetric about c and may include endpoints (must test separately).
• Ratio Test (Primary Tool): Used to find R via lim |a?(x – c)¹ / a?(x – c)?| < 1; simplifies to |x – c| < lim |a? / a?| when the limit exists.
• Endpoint Behavior: Convergence at x = c ± R is not guaranteed by the Ratio Test; must test each endpoint using other tests (e.g., p-series, Alternating Series Test).

Core Questions

WHAT (definitional)

Q: What is a power series centered at c? A: A series of the form ? a?(x – c)?, where a? are coefficients and c is the center. Trap/Clarification: The series is not a polynomial; it’s an infinite sum, so convergence depends on x.

Q: What does the radius of convergence R represent? A: The maximum distance |x – c| from the center c for which the series converges absolutely. Trap/Clarification: R is not the interval length; the interval is 2R wide (excluding endpoints).

WHY (causal/explanatory)

Q: Why is the Ratio Test the go-to method for finding R? A: It directly yields lim |a? / a?| = L, so R = 1/L (if L-0 or ?), simplifying the convergence condition |x – c| < R. Trap/Clarification: The Ratio Test fails if the limit L doesn’t exist (e.g., for series with factorials or alternating signs); use Root Test instead.

Q: Why must endpoints be tested separately? A: The Ratio Test gives no information about convergence at x = c ± R; other tests (e.g., p-series, AST) are needed. Trap/Clarification: Assuming endpoints converge because R is finite is a common error—always test!

HOW (process/application)

Q: How do you find the radius of convergence R? A: Apply the Ratio Test to ? a?(x – c)?, compute lim |a?(x – c) / a?| = |x – c| · L, and set |x – c| < 1/L = R. Trap/Clarification: If L = 0, R = ?; if L = ?, R = 0 (converges only at x = c).

Q: How do you determine the interval of convergence? A: 1) Find R via Ratio/Root Test, 2) write the open interval (c – R, c + R), 3) test endpoints x = c ± R separately. Trap/Clarification: Forgetting to test endpoints is a top exam mistake—the IoC may be open, closed, or half-open.

CAN (conditions/possibilities)

Q: Can a power series converge at x = c but nowhere else? A: Yes, if R = 0 (e.g., ? n! x? converges only at x = 0). Trap/Clarification: This is not the same as divergence; the series converges trivially at its center.

Q: Under what conditions does a power series converge uniformly? A: On any closed subinterval [-a, a]? (c – R, c + R); uniform convergence fails at endpoints unless the series converges there. Trap/Clarification: Uniform convergence ? absolute convergence—test endpoints for both!

Quick Facts & Traps

• Fact: R is always ? 0 and can be 0, ?, or a positive real number; the IoC is never empty (always includes x = c).
• Trap: Assuming R = 1 for all series-Reality: R depends on coefficients (e.g., ? x?/n! has R = ?).
• Fact: The IoC is symmetric about c but may exclude one or both endpoints (e.g., ? x?/n converges on [-1, 1)).
• Trap: Using the Ratio Test on endpoints-Reality: The Ratio Test always fails at x = c ± R (limit = 1).
• Fact: If ? a?(x – c)? converges at x = d, then it converges absolutely for all x with |x – c| < |d – c|.
• Trap: Misapplying the Root Test by ignoring |x – c|-Reality: The Root Test gives lim |a?|¹/? · |x – c| < 1, so R = 1/lim |a?|¹/?.

Rapid-Fire True/False

• Statement: If a power series converges at x = 5, it must converge at x = 3. Answer: TRUE Why the common mistake happens: Students forget the absolute convergence property within the radius.

• Statement: The interval of convergence for ? (x – 2)?/n² is [1, 3]. Answer: TRUE Why the common mistake happens: Students test endpoints but misapply the p-series test (forgets n²-p = 2 > 1).

• Statement: A power series with R = 2 must converge at x = 0 and x = 4. Answer: FALSE Why the common mistake happens: Students assume endpoints are included without testing (e.g., ? (x – 2)?/n diverges at x = 4).