Practice a College Board-style free response question on power series, the ratio test, interval of convergence, and Maclaurin series. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 10 · Infinite Sequences & Series (BC)
Consider the power series Σ (n=1..∞) xⁿ / (n·3ⁿ).
Also let f be the function with Maclaurin series f(x) = Σ (n=0..∞) xⁿ/n!.
A
Use the ratio test to find the radius of convergence of Σ xⁿ/(n·3ⁿ).
✓ Model answer
Ratio: |aₙ₊₁/aₙ| = |xⁿ⁺¹/((n+1)3ⁿ⁺¹) · (n·3ⁿ)/xⁿ| = |x|/3 · n/(n+1). As n → ∞, this → |x|/3. Convergence requires |x|/3 < 1, i.e. |x| < 3, so the radius of convergence is R = 3.
Why it scores: Sets up the ratio, simplifies to |x|/3, and concludes R = 3. An algebra slip in the ratio, or forgetting the limit, loses points.
B
Determine the interval of convergence of Σ xⁿ/(n·3ⁿ), testing both endpoints.
✓ Model answer
From R = 3, check x = ±3. At x = 3: Σ 3ⁿ/(n·3ⁿ) = Σ 1/n, the harmonic series, which diverges. At x = −3: Σ (−3)ⁿ/(n·3ⁿ) = Σ (−1)ⁿ/n, the alternating harmonic series, which converges. So the interval of convergence is [−3, 3).
Why it scores: Tests both endpoints, correctly identifies harmonic (diverges) and alternating harmonic (converges), and writes [−3, 3). Skipping an endpoint loses credit.
C
The function f(x) = Σ xⁿ/n! is eˣ. Write the first four nonzero terms of the Maclaurin series for f, and use them to approximate f(1) = e.
✓ Model answer
The first four nonzero terms are 1 + x + x²/2! + x³/3!. At x = 1: 1 + 1 + 1/2 + 1/6 = 8/3 ≈ 2.667, an approximation of e ≈ 2.718.
Why it scores: Lists the correct terms 1 + x + x²/2 + x³/6 and evaluates at x = 1 to 8/3. Dropping a factorial or a term loses the point.
How to score points on AP Calculus FRQs
Use the ratio test to get the radius, then test both endpoints for the interval.
Name the series at the endpoints (harmonic, alternating harmonic, p-series) to justify convergence.
Know the standard Maclaurin series for eˣ, sin x, cos x, and 1/(1−x) cold.
Keep the factorials in Taylor coefficients.
Answer the verb. "Find" wants the value; "determine the interval" needs the endpoint work.