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Radius of Convergence Calculator

Enter aₙ as a formula in n or as a list of coefficients and get R by the ratio or root test — plus the endpoint behaviour most calculators skip: Σ(−1)ⁿ/(n+1) converges at x = 1 and diverges at x = −1.

Related tools

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Series Convergence Test CalculatorPick a series and see which test settles it — p-series, ratio, root, Leibniz, Cauchy condensation, limit comparison — with the limit it produces and whether the convergence is absolute or conditional.Antilog Calculator (inverse logarithm)The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.Bernoulli ODE SolverSolves y′ + P(x)y = Q(x)yⁿ by showing the substitution v = y^(1−n) that turns it into a linear equation, then integrating. The two degenerate cases are handled explicitly: n = 0 is already linear, n = 1 is separable.Convolution CalculatorLinear and circular discrete convolution with the sliding-and-summing steps shown term by term, plus a symbolic continuous convolution solved exactly through the Laplace convolution theorem.Derivative CalculatorCompute single, partial, implicit and directional derivatives symbolically.Direction Field / Slope Field PlotterDraws f(x, y) as a grid of tangent dashes, so the shape of every solution is visible before a single one is computed. Add initial conditions and each spawns an RK4 curve that stays tangent to the field.Directional Derivative CalculatorCompute the directional derivative Dûf = ∇f · û at a point along a direction.Double Integral CalculatorNumerically evaluate a double integral ∬ f(x,y) over a rectangular region.

Radius of Convergence Calculator works straight from this page — free, instant, nothing to install. It sits under Calculus in our catalogue, alongside Series Convergence Test Calculator and Antilog Calculator (inverse logarithm).

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Radius of Convergence Calculator do?

Enter aₙ as a formula in n or as a list of coefficients and get R by the ratio or root test — plus the endpoint behaviour most calculators skip: Σ(−1)ⁿ/(n+1) converges at x = 1 and diverges at x = −1.

When would I actually use this?

Checking a derivative or an integral you worked out by hand, finding where a function turns, and getting a numeric answer when no closed form exists.

What is the most common mistake?

Integrating across a discontinuity as if it were not there. A numeric method will happily return a finite value for an integral that diverges — check the domain before trusting the number.

How is Radius of Convergence Calculator different from Series Convergence Test Calculator?

They sit next to each other but answer different questions: Series Convergence Test Calculator is the one to open when you need it to pick a series and see which test settles it — p-series, ratio, root, Leibniz, Cauchy condensation, limit comparison — with the limit it produces and whether the convergence is absolute or conditional. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Antilog Calculator (inverse logarithm) is the closest one after this: The antilogarithm undoes the logarithm: antilog_b(y) = bʸ, so log_b(bʸ) = y. Pick base 10, e, 2 or your own, enter the exponent, and the result comes back with the inverse relationship spelled out and the round trip checked. Bases of zero or less, and base 1, are rejected with the reason.

What else is worth having open alongside it?

Bernoulli ODE Solver and Convolution Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Symbolic results are exact; numeric ones come from adaptive quadrature or a standard step method, and the tool reports which. Where both are available, compare them — a large gap means the problem is ill-conditioned.

Further reading

All guides →
ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.ExplainerWhat Is a Z-Score? Standard Deviations Above the MeanA z-score is z = (x − μ) / σ — how many standard deviations a value sits from the mean. Learn how to compute it, read the standard normal curve, and turn it into a percentile.How-toHow to Find the Distance Between Two PointsUse the distance formula d = √((x₂−x₁)² + (y₂−y₁)²) to measure the straight-line distance between two points on a plane, with a worked example and the midpoint too.How-toHow to Calculate the Slope of a Line: Rise Over RunFind the slope of a line from two points with m = (y₂ − y₁) / (x₂ − x₁), and read what a positive, negative, zero, or undefined slope tells you.