Education
Convergence Calculator
Last updated: May 31, 2026
Written by Blake Boege
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.
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) |
|---|---|
| 10 | 0.630957 |
| 50 | 0.855148 |
| 100 | 0.912011 |
| 150 | 0.935374 |
Conclusion via Root Test:
Converges
The Root Test limit L = lim |a_n|^(1/n) ≈ 0.9354 which is less than 1.
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).
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 → ∞) |a_n₊₁ / a_n|
Root Test Limit
L = lim (n → ∞) |a_n|¹/ⁿ
Absolute vs. Conditional Convergence
A series $\sum a_n$ converges absolutely if the series of absolute values $\sum |a_n|$ converges. If $\sum a_n$ converges but $\sum |a_n|$ diverges (such as the alternating harmonic series $\sum (-1)^n / n$), 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.
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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 |a_{n+1} / a_n|. 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 |a_n|^(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 a_n as n approaches infinity is not equal to 0, then the series must diverge. If the limit is 0, the test is inconclusive.
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