Interval of Convergence Calculator

What Is an Interval of Convergence?

The interval of convergence of a power series is the complete set of real numbers x for which the series produces a finite (convergent) sum. A power series takes the general form:

where c is the center of the series and a_n are the coefficients. The convergence interval is always symmetric about c, forming the open interval (c−R, c+R) where R is the radius of convergence. Endpoint behavior must be tested separately using additional convergence tests.

Understanding the Radius of Convergence

The radius of convergence R determines how far from the center the series converges absolutely. There are exactly three cases:

• R = 0: The series converges only at the single point x = c and diverges everywhere else.
• R = ∞: The series converges for every real number, giving the interval (−∞, +∞).
• 0 < R < ∞: The series converges on the open interval (c−R, c+R) and diverges outside. Each endpoint must be tested individually using other convergence criteria.

The Ratio Test for Radius of Convergence

The Ratio Test is the most widely applied method for finding R in a Calculus II course. For a power series ∑a_n(x−c)ⁿ, apply the limit:

The series converges absolutely when L·|x−c| < 1, which yields the radius R = 1/L. Special cases: if L = 0 then R = ∞; if L = ∞ then R = 0. When L = 1, the Ratio Test is inconclusive and the Root Test or another method must be used.

The Root Test (Cauchy-Hadamard Formula)

The Root Test — also called the Cauchy-Hadamard theorem — is especially useful when series terms involve exponentials or high powers:

This formula directly yields the radius of convergence. Where both the Ratio Test and Root Test are applicable, they always produce the same value of R. The Root Test is preferred when the Ratio Test leads to indeterminate forms.

Endpoint Analysis: The Critical Step

The Ratio and Root Tests are inconclusive at the boundary points x = c±R (where the tests yield exactly L = 1). Each endpoint must be evaluated independently by substituting that x-value into the original series and applying appropriate tests:

• p-Series Test: ∑1/n^p converges if p > 1, diverges if p ≤ 1.
• Alternating Series Test: If terms alternate in sign, decrease in magnitude, and approach zero, the series converges.
• Comparison Test: Bound the series above or below by a known convergent or divergent series.
• Divergence Test: If terms do not approach zero, the series must diverge.

Open vs. Closed Intervals in Convergence Notation

The final interval of convergence uses standard interval notation where square brackets indicate inclusion and round parentheses indicate exclusion of the endpoint:

• (c−R, c+R) — Open interval: both endpoints diverge.
• [c−R, c+R) — Half-open: left endpoint converges, right diverges.
• (c−R, c+R] — Half-open: right endpoint converges, left diverges.
• [c−R, c+R] — Closed interval: both endpoints converge.

Mixed intervals arise frequently — for example, the alternating series ∑(−1)ⁿxⁿ/n converges conditionally at one endpoint but diverges at the other.

Absolute vs. Conditional Convergence

Inside the open interval (c−R, c+R), every power series converges absolutely — meaning both the original series and the series of absolute values converge. At the endpoints, the series may converge only conditionally, where the series converges but ∑|a_n(x−c)ⁿ| does not.

The alternating harmonic series ∑(−1)ⁿ⁺¹/n is the canonical example of conditional convergence. It sums to ln(2) but its absolute-value counterpart — the harmonic series — diverges.

Engineering and Applied Science Applications

Power series convergence underpins a wide range of engineering and scientific computations:

• Taylor and Maclaurin Series: The Taylor expansion of e^x, sin(x), cos(x), and ln(1+x) each converge on specific intervals essential for approximation in numerical computing.
• Signal Processing: The Z-transform uses power series with a region of convergence analogous to the interval of convergence.
• Control Theory: Transfer functions can be expanded as power series; instability arises if operating outside the convergence radius.
• Perturbation Methods: In quantum mechanics and fluid dynamics, perturbation series must remain within their convergence radius for physical validity.
• Numerical ODE Solvers: Power series methods for differential equations require careful convergence radius estimation to bound errors.

Worked Example: Full Solution

Problem: Find the interval of convergence for ∑_n=0^∞ xⁿ/3ⁿ.

Step 1 — Apply the Ratio Test: Compute lim|a_n+1/a_n| = lim|xⁿ⁺¹/3ⁿ⁺¹ · 3ⁿ/xⁿ| = |x|/3. The series converges when |x|/3 < 1, i.e. |x| < 3. So R = 3 and c = 0.

Step 2 — Open Interval: (−3, 3).

Step 3 — Test x = −3: The series becomes ∑(−1)ⁿ. Since the terms do not approach 0, the series diverges by the Divergence Test. Left endpoint: excluded.

Step 4 — Test x = 3: The series becomes ∑1ⁿ = ∑1, which diverges. Right endpoint: excluded.

Conclusion: Both endpoints diverge. The interval of convergence is (−3, 3) — an open interval.