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Series Convergence Test Calculator

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.

Related tools

All Sequences & series tools →
Infinite Series Sum CalculatorTen classic families — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, Σ1/n! — with their closed form (π²/6, ln 2, π/4, e), the convergence condition and the partial sums.Power Series CalculatorExpand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.Radius of Convergence CalculatorEnter 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.Fourier Series Coefficients Calculatora₀, aₙ and bₙ over any period, computed exactly by integration by parts when f is a polynomial and by Gauss–Legendre quadrature otherwise, with the truncated series plotted against f so the Gibbs overshoot is visible.Maclaurin Series CalculatorThe Taylor series at 0 for twelve standard functions — sin, cos, eˣ, ln(1+x), √(1+x), arcsin, arctan and more — with exact fractions such as 1/120, the general term and the error at a chosen x.Taylor Series CalculatorType any f(x) — tan(x), sqrt(1+x), ln(1+x), exp(x)/cos(x) — pick the expansion point and get exact rational coefficients up to order 20, the derivatives f⁽ⁿ⁾(a) and the detected general term.Chi-Square Test Calculatorχ² test of independence on any contingency table, showing observed, expected, the contribution and the standardised residual of every cell, plus Cramér's V. Cells with an expected count below 5 are flagged, because that is when the test stops being valid.Divisibility Test CalculatorTest a number against the divisors 2 to 12, the primes up to 100 or your own list — and see the mental rule and the working, not just yes or no.

Series Convergence Test Calculator is free to use as often as you like, directly from this page. It sits under Sequences & series in our catalogue, alongside Infinite Series Sum Calculator and Power Series Calculator.

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 Series Convergence Test Calculator do?

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.

When would I actually use this?

Finding the nth term without listing every one before it, summing a long run in one step, and recognising which family a sequence belongs to.

What is the most common mistake?

Starting the index at the wrong end. Whether the first term is a₀ or a₁ shifts every result by one position, and the two conventions are both common.

How is Series Convergence Test Calculator different from Infinite Series Sum Calculator?

They sit next to each other but answer different questions: Infinite Series Sum Calculator is the one to open when you need it to ten classic families — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, Σ1/n! — with their closed form (π²/6, ln 2, π/4, e), the convergence condition and the partial sums. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Power Series Calculator is the closest one after this: Expand eˣ, sin, cos, ln, √x, arctan and six more about any centre a, with exact fractional coefficients, the general term, and the interval of convergence including both endpoints.

What else is worth having open alongside it?

Radius of Convergence Calculator and Fourier Series Coefficients Calculator — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The closed forms are the standard ones, and the tool states the indexing convention it uses so a result can be compared against a textbook without ambiguity.

Further reading

All guides →
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.ExplainerResistors in Series and in Parallel: The Formulas, and the Three Things They Do Not Tell YouSeries adds, parallel adds reciprocals — that part is easy. The parts that catch people: which member dominates the answer, why two 5 % resistors in parallel are still 5 %, and why the smallest resistor in a parallel pair takes the largest share of the power.How-toHow to Calculate an LED Series ResistorSize the series resistor for any LED with one formula: R = (Vs − Vf) / I. Follow the steps, see a worked 5 V example, and pick the right resistor.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.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.