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Calculator Answers

Convergence Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A convergence calculator is an educational utility that tests infinite numerical series to determine if they converge or diverge. It supports several classical tests, including the Ratio Test and Root Test, by calculating their limits at large indices. In auto mode, the calculator tries multiple tests sequentially and falls back to a numerical partial sum stabilization check to simulate Comparison and Integral Tests. Students and educators use this solver to analyze series convergence behaviors and verify homework calculations with step-by-step limits.

Test infinite series for convergence or divergence. Run Ratio, Root, Comparison, or Integral tests with numerical step explanations.

See also: Interval of Convergence Calculator, Radius of Convergence Calculator, and Cone Volume Calculator.

Quick Answer

Determine if an infinite series converges or diverges. Enter the general term a_n and choose a test to view the limit conclusion.

Test Type

Try Examples

Numerical Values evaluated at n

n|a_n|^(1/n)
100.630957
500.855148
1000.912011
1500.935374
Convergence Test Results

Conclusion via Root Test:

Converges

The Root Test limit L = lim |a_n|^(1/n) ≈ 0.9354 which is less than 1.

Term1/n²
Test AppliedRoot Test
Resulting StateConverges

Notice: This calculator uses numerical limits and sum stabilization heuristics to determine convergence.

For academic rigor, verify results using analytical convergence proofs (e.g. Ratio, Root, Comparison, or Integral tests).

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Examples

1/n² (converges)

Ratio Test inconclusive (L=1); sum stabilizes (S_1000 ≈ 1.643)

1/n (diverges)

Diverges; n-th term approaches 0 but sum grows without bound (harmonic series)

n! / n^n (converges)

Ratio Test limit L ≈ 0.368 < 1 (converges)

3^n / (n * 2^n)

Ratio Test limit L = 1.5 > 1 (diverges)

How it works

The calculator evaluates successive terms of the series at large indices (n = 10, 50, 100, 150) and computes limits or sums:

Ratio Test Limit

L = lim (n → ∞) |an+1 / an|

Root Test Limit

L = lim (n → ∞) |an|1/n

Absolute vs. Conditional Convergence

A series Σ an converges absolutely if the series of absolute values Σ |an| converges. If Σ an converges but Σ |an| diverges (such as the alternating harmonic series Σ (−1)n / n), the series is said to converge conditionally. Absolute convergence is stronger because absolutely convergent series can be rearranged in any order without changing their sum.

Related Mathematical Solvers

If your series contains a variable x (a power series), visit our Radius of Convergence Calculator or the Interval of Convergence Calculator.

Related Calculators

More tools from Education

Interval of Convergence CalculatorFind the interval of convergence of a power series including endpoint convergence testing using partial sum heuristics.Radius of Convergence CalculatorCalculate the radius of convergence R of any power series using the Ratio Test with numerical limit approximations.Cone Volume CalculatorCalculate the volume of a cone using the formula V = (1/3)πr²h with radius and height inputs.Polynomial CalculatorPerform polynomial long division, find quotient and remainder, and work with polynomial expressions step by step.

Frequently asked questions

An infinite series converges if the sequence of its partial sums approaches a specific, finite limit. If the partial sums grow without bound or oscillate infinitely, the series diverges.

The Ratio Test calculates the limit L = lim |an+1 / an|. If L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive.

The Root Test calculates the limit L = lim |an|1/n. Like the Ratio Test, if L < 1 the series converges, if L > 1 it diverges, and if L = 1 it is inconclusive.

The Divergence Test (or n-th term test) states that if the limit of the terms an as n approaches infinity is not equal to 0, then the series must diverge. If the limit is 0, the test is inconclusive.