Radius and Interval of Convergence of Power Series

Historical Context & Motivation

The idea of representing functions as infinite sums of powers dates back centuries, long before the formal machinery of convergence was established. Early mathematicians such as Isaac Newton and James Gregory manipulated power series freely, expanding functions like sin x and ln(1 + x) into infinite polynomials and using them for computation. These expansions yielded remarkably accurate numerical results within certain ranges, yet produced wildly divergent nonsense outside those ranges. The central question that eventually crystallized was deceptively simple: for which values of x does a given power series actually converge to a finite sum? Answering that question rigorously required new mathematical ideas about limits, absolute convergence, and careful endpoint analysis — ideas that took over a century to mature.

The fundamental gap these mathematicians addressed is one you will master in this lesson: given a power series Σ cₙ(x − a)ⁿ, how do we determine the precise set of x-values for which the series converges? The answer splits naturally into two tasks — finding the radius of convergence R, and then checking the endpoints to pin down the full interval of convergence. This two-step process is a cornerstone of the AP Calculus BC exam.

Core Principles & Definitions

A power series centered at a has the general form Σ cₙ(x − a)ⁿ for n = 0, 1, 2, …, where the coefficients cₙ are constants and a is the center of the series. The series always converges at x = a itself (every term after the zeroth is zero), but the crucial question is how far from a the convergence extends. A celebrated theorem guarantees that exactly one of three scenarios holds: the series converges only at x = a, it converges for all real x, or there exists a positive number R such that the series converges absolutely for |x − a| < R and diverges for |x − a| > R. That number R is the radius of convergence.

Visualizing the Interval of Convergence

The following diagram illustrates the anatomy of a power series interval of convergence on the real number line. The center a sits at the midpoint, and the radius R extends symmetrically in both directions. Within the open interval (a − R, a + R), absolute convergence is guaranteed. Outside, the series diverges. The two endpoints require individual investigation and may independently converge or diverge, yielding four possible interval shapes.

Notice the perfect symmetry about the center a: the open interval of absolute convergence always has equal half-width R on both sides. This symmetry is a consequence of the Ratio and Root Tests depending only on |x − a|, which measures distance from the center regardless of direction. However, the endpoints break this symmetry — one may converge while the other diverges — because substituting x = a − R and x = a + R into the series can produce structurally different numerical series (e.g., an alternating series on one side but a divergent p-series on the other).

Mathematical Framework

The two primary tools for computing the radius of convergence are the Ratio Test and the Root Test. Both are applied to the general term of the power series and produce a limit that depends on |x − a|. Setting that limit less than 1 yields absolute convergence, and solving for x identifies the open interval (a − R, a + R). In nearly every AP Calculus BC problem, the Ratio Test is the method of choice.

Step-by-Step Procedure

1. Step 1: Identify the general term aₙ = cₙ(x − a)ⁿ and write out aₙ₊₁.
2. Step 2: Compute lim (n→∞) |aₙ₊₁ / aₙ| and simplify. The result will typically be |x − a| multiplied by a constant.
3. Step 3: Set the limit < 1 and solve for x to find the open interval (a − R, a + R). Extract R.
4. Step 4: Test x = a − R: substitute into the original series and use an appropriate convergence test (p-series, AST, comparison, etc.).
5. Step 5: Test x = a + R: substitute and test convergence separately.
6. Step 6: State the interval of convergence using the correct bracket/parenthesis notation based on endpoint results.

Endpoint Analysis & Classification

The most common source of errors on the AP exam is neglecting the endpoint analysis or applying the wrong test at each endpoint. Once R has been found, substituting x = a + R into Σ cₙ(x − a)ⁿ gives Σ cₙRⁿ, and substituting x = a − R gives Σ cₙ(−R)ⁿ = Σ (−1)ⁿcₙRⁿ. These are ordinary numerical series (no x remaining), and you must test each with standard convergence tools. The following diagram and table summarize the most frequently encountered endpoint series types and the corresponding tests.

Worked Example

Let us find the radius and interval of convergence of the power series Σ (n=1 to ∞) (x − 3)ⁿ / (n × 2ⁿ). This example is representative of the difficulty level you will encounter on the AP Calculus BC exam.

Common Pitfalls & Comparisons

Students frequently lose points on the AP exam not because they cannot apply the Ratio Test, but because of avoidable procedural mistakes. Understanding these common pitfalls — and how they contrast with correct practice — will significantly improve your accuracy.

Connection to Taylor Series & Advanced Theory

The radius and interval of convergence are not isolated technical curiosities — they sit at the heart of the theory connecting power series to the functions they represent. Every Taylor series is a power series whose coefficients are derived from the derivatives of a function at a center: cₙ = f⁽ⁿ⁾(a)/n!. The radius of convergence then tells you the largest open interval on which the Taylor series actually equals the original function. Understanding R thus becomes the gateway to using series for approximation, integration of non-elementary functions, and solving differential equations — all key topics in BC Calculus and beyond.

A particularly elegant fact — explored in complex analysis — explains why R has the value it does. Consider f(x) = 1/(1 + x²). Its Taylor series centered at 0 is Σ (−1)ⁿx²ⁿ with R = 1, even though f(x) is perfectly smooth for all real x. The mystery resolves in the complex plane: f(z) has singularities at z = ±i, which are exactly distance 1 from the center z = 0. The radius of convergence always equals the distance to the nearest complex singularity. While this is beyond the AP exam, knowing it provides deep intuition for why certain "well-behaved" functions have finite R.

Practice Problems

Summary & Review