Interval of Convergence Calculator

What This Calculator Determines

This tool turns the values you already found — a center, a radius, and the convergence status of each endpoint — into a complete, correctly bracketed interval of convergence. It does not analyze a series expression on its own. You supply the radius (from the Ratio Test or Root Test, covered below) and the endpoint results, found by substituting each boundary value into the original series and testing it separately. From there, the calculator builds the final interval, classifies the type of convergence, and lays out the reasoning step by step.

Each field on the form corresponds to a specific part of that process:

• Center (c): the point the series is expanded around — for example, 0.
• Radius (R): the distance from the center where the series converges absolutely, found with the Ratio or Root Test. Must be 0 or greater.
• Left Endpoint Status: whether the series converges when x = c − R, based on your own testing.
• Right Endpoint Status: whether the series converges when x = c + R, based on your own testing.
• Notes (optional): space to record your reasoning. Anything entered here carries over into the copied summary and the PDF report.

How to Use This Calculator

Follow these steps to go from a series to a verified interval of convergence:

1. Find the radius of convergence R using the Ratio Test or Root Test. The sections below walk through both methods, and our guide on How to Find the Interval of Convergence for a Series covers the full process with more examples.
2. Enter the center c and the radius R into the form above. R must be 0 or greater.
3. Substitute x = c − R into the original series and test it with the p-Series Test, Alternating Series Test, Comparison Test, or Divergence Test — whichever applies. Choose Converges, Diverges, or Unknown for the left endpoint based on what you find.
4. Repeat the same process for the right endpoint at x = c + R.
5. Add notes if you want a written record of your reasoning. This step is optional.
6. Click Calculate Interval. The results include your metrics, the final interval, a step-by-step writeup, and a summary card for each endpoint.
7. Click Copy Result for a plain-text summary, or Download PDF for a formatted report. Click Reset to clear the form and erase any saved progress stored in your browser.

A few cases fall outside this normal flow. If R = 0, the series converges only at the center, and the endpoint dropdowns have no effect on the answer — any selection satisfies the form. If a series converges for every real number (R = ∞, as with the series for eˣ or sin x), this calculator is not the right tool, since there are no finite endpoints to test; the interval is simply (−∞, ∞). And choosing Unknown for an endpoint does not pause the calculation — it still produces an interval with an excluded bracket at that point, but labels the result Partially Determined Convergence as a reminder that the boundary still needs proper testing.

Understanding Your Results

After you click Calculate Interval, the results panel breaks your answer into several parts:

• Open Interval: the interval before endpoints are considered. It always appears in parentheses, since endpoints are decided separately.
• Final Interval: the complete answer, with brackets matched to what you selected for each endpoint.
• Convergence Type: labels the result as open, closed, half-open, single-point, or partially determined.
• Step-by-Step Solution: the full reasoning, from your parameters through the boundary points to the final bracket choice.
• Endpoint Summary: two cards showing the status and bracket for x = c − R and x = c + R side by side.

Limitations of This Calculator

This tool is built to format and check an answer you have already worked out, not to replace that analysis. Keep these boundaries in mind:

• It works with real-valued power series in the standard ∑aₙ(x−c)ⁿ form and a single, finite radius. It does not apply to complex-valued series, Fourier or Laurent series, or series with more than one radius of convergence in different directions.
• It does not verify that your endpoint testing is mathematically correct. It takes your Converges/Diverges/Unknown selection at face value, so an incorrect choice produces a wrong, but confidently formatted, final interval.
• It is not designed for R = ∞. Series that converge everywhere should simply be written as (−∞, ∞) directly, without running them through the endpoint logic here.
• Very long decimal inputs are rounded for display, so an irrational or repeating value may appear as a rounded approximation rather than an exact figure.
• Saved inputs live only in this browser's local storage. They will not follow you to a different device or browser, clearing your browser data removes them, and Reset erases them permanently.

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.

Common Mistakes with Interval of Convergence

A handful of errors show up often enough in Calculus II coursework to call out directly:

• Treating the open interval as the final answer. The Ratio Test only proves absolute convergence inside (c−R, c+R) — each endpoint still needs its own test.
• Reapplying the Ratio Test at an endpoint. Because R is defined so that L = 1 exactly at x = c±R, the Ratio Test is inconclusive there by construction. Substitute the endpoint value into the original series and use the p-Series, Alternating Series, Comparison, or Divergence Test instead.
• Assuming both endpoints behave the same way. Many series, including the ln(1+x) expansion, converge at one endpoint and diverge at the other, producing a half-open interval rather than a symmetric one.
• Dropping the absolute value when computing the Ratio Test limit. Without it, the computed radius or its sign can come out wrong.
• Treating the radius and the interval as the same thing. R is a single non-negative number measuring distance from the center; the interval is the full set of x-values, written with brackets or parentheses. Our guide on Radius vs. Interval of Convergence: What Is the Difference? looks at this distinction in more detail.

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.