In AP Calculus BC, the radius of convergence R is the distance from the center of a power series within which the series is guaranteed to converge. The series converges for |x - r| < R, diverges for |x - r| > R, and the endpoints must be tested separately. You find R with the ratio test.
A power series has the form ∑aₙ(x - r)ⁿ, where r is the center. The radius of convergence, usually called R, tells you how far you can move away from that center in either direction and still have the series converge. Picture the center as home base. Inside a distance of R from home base, the series converges. Outside that distance, it diverges. Exactly at distance R (the two endpoints), anything can happen, so you have to check those points one at a time.
The CED spells out the three possibilities. If R = 0, the series converges only at its center and nowhere else. If R is a positive finite number, the series converges on an open interval (r - R, r + R), plus possibly one or both endpoints. If R is infinite, the series converges for every real number x. The standard tool for finding R is the ratio test, since that test naturally produces an inequality like |x - r| < R when you require the limit of the ratio to be less than 1.
Radius of convergence lives in Topic 10.13 of Unit 10 (Infinite Sequences and Series), which is BC-only. It directly supports learning objective 10.13.A, determining the radius and interval of convergence of a power series. It also matters beyond 10.13. Every Taylor and Maclaurin series you build later in Unit 10 is a power series, so its radius of convergence tells you where that series actually equals the function it represents. When the College Board says a Maclaurin series 'converges to f(x) for all x in the interval of convergence,' the radius is what defines that region. This is one of the most reliable point-earners on the BC exam, because the procedure (ratio test, solve the inequality, test endpoints) is the same every time.
Keep studying AP Calculus Unit 10
Visual cheatsheet
view galleryInterval of Convergence (Unit 10)
The radius gives you the size of the convergence region; the interval gives you the actual set of x-values. R = 3 centered at 0 means the open interval (-3, 3) is guaranteed, but whether the interval is open, closed, or half-open depends on testing x = -3 and x = 3 individually. Radius first, then endpoints, then interval.
Ratio Test (Unit 10)
The ratio test from Topic 10.8 is how you actually compute R. You take the limit of |aₙ₊₁(x-r)ⁿ⁺¹ / aₙ(x-r)ⁿ|, set it less than 1, and solve for |x - r|. The number on the right side of that inequality is your radius. The ratio test is inconclusive when the limit equals 1, which is exactly why endpoints need separate checks.
Taylor and Maclaurin Series (Unit 10)
Topics 10.14 and 10.15 build power series representations of functions like eˣ, sin x, and 1/(1-x). The radius of convergence tells you where the series and the function actually agree. The series for eˣ has R = ∞, but the geometric-style series for 1/(1-x) only works with R = 1. Same procedure, very different regions.
Divergence Test (Unit 10)
The nth term test is often the fastest tool for endpoint checks. If the terms of the series at an endpoint don't go to zero, the series diverges there and that endpoint stays out of the interval of convergence.
This shows up two ways. In multiple choice, you'll compute R for a given power series using the ratio test, or interpret what R means. Practice questions hit exactly these angles, like what R = ∞ means (converges for all real x), what R = 0 means (converges only at the center), and what an R of 3 centered at 0 tells you about the interval (the open interval (-3, 3) is guaranteed, endpoints unknown until tested). A sneaky favorite is the polynomial, like f(x) = 4x² + 3x - 2. A polynomial is a power series with finitely many nonzero terms, so it converges everywhere and R = ∞. On FRQs, radius of convergence anchors the Unit 10 series question that closes out the BC exam almost every year. The 2024 BC FRQ Q6 gave a Maclaurin series and asked you to work with its radius of convergence, and the 2025 BC FRQ Q6 did the same with a Taylor series centered at x = 4. Expect to show the ratio test setup, the limit, and the inequality work for full credit, not just the final number.
The radius is a single number (a distance); the interval is a set of x-values. If a series is centered at 2 with R = 3, the radius is 3 but the interval is some version of (-1, 5), possibly including one or both endpoints. Students lose points by stopping at the radius when the question asks for the interval. The radius gets you the open interval for free, but the interval of convergence requires plugging both endpoints back into the series and testing each with a convergence test.
The radius of convergence R is the distance from the center of a power series within which the series is guaranteed to converge.
Use the ratio test to find R by setting the limit of the ratio of consecutive terms less than 1 and solving for |x - r|.
If R = 0 the series converges only at its center, and if R = ∞ it converges for every real number x.
The radius gives you the open interval (r - R, r + R), but you must test both endpoints separately to find the full interval of convergence.
The ratio test is inconclusive at the endpoints, which is exactly why endpoint behavior can differ from the inside of the interval.
A polynomial is a power series with finitely many terms, so its radius of convergence is automatically infinite.
It's the distance R from the center of the series within which the series converges. For a series centered at r, convergence is guaranteed for all x with |x - r| < R, and divergence is guaranteed for |x - r| > R.
No. The radius is one number, while the interval is the actual set of x-values where the series converges. R = 3 centered at 0 guarantees the open interval (-3, 3), but the final interval might be (-3, 3), [-3, 3), (-3, 3], or [-3, 3] depending on endpoint tests.
The series converges for every real number x. Classic examples are the Maclaurin series for eˣ, sin x, and cos x, plus any polynomial.
No, it still converges at exactly one point, its center. Every power series converges at x = r because all the terms after the constant become zero there. R = 0 just means that center is the only place it works.
Apply the ratio test. Compute the limit of |aₙ₊₁/aₙ| times |x - r|, require it to be less than 1, and solve for |x - r| < R. On FRQs like 2024 and 2025 BC Q6, you need to show the limit setup and inequality work, not just state R.
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