10.13 Radius and Interval of Convergence of Power Series

Finding the Radius of Convergence

Use the Ratio Test on a power series ∑cₙ(x−a)ⁿ: compute limₙ→∞|cₙ₊₁(x−a)ⁿ⁺¹ / cₙ(x−a)ⁿ| and set it < 1. This gives |x−a| < R, the radius of convergence. If the limit is always 0, R = ∞ (converges everywhere); if it's always infinite except at x = a, R = 0 (converges only at x = a).

Finding the Interval of Convergence

The radius R gives an open interval (a−R, a+R) where the series converges absolutely. To find the full interval of convergence, you MUST separately test both endpoints x = a−R and x = a+R using other tests (often the Alternating Series Test or Limit Comparison Test), since the Ratio Test is inconclusive there.

Endpoint Behavior Matters

Each endpoint can independently converge or diverge, giving four possible interval types: (a−R, a+R), [a−R, a+R), (a−R, a+R], or [a−R, a+R]. Always plug the endpoint value into the original series (not the limit expression) and analyze the resulting numerical series for convergence.

Key terms

Radius of convergence (R)

The value such that a power series converges for |x−a| < R and diverges for |x−a| > R; found via the Ratio Test.

Interval of convergence

The complete set of x-values where the power series converges, including any endpoints that test as convergent.

Why test endpoints separately

The Ratio Test gives a limit of exactly 1 at the endpoints, which is inconclusive, so another test must be used there.

R = ∞ case

The power series converges for all real x; the Ratio Test limit is 0 regardless of x.

R = 0 case

The power series converges only at the center x = a; the Ratio Test limit is infinite for any x ≠ a.

Try an AP-style question

What is the interval of convergence of Σ xⁿ/n?

1. (−1, 1]
2. (−1, 1)
3. [−1, 1)
4. [−1, 1]