Convergence Calculator
Check whether common infinite sequences and series converge or diverge using geometric-series, p-series, ratio-test, and nth-term criteria.
Select geometric, p-series, ratio test, or nth-term test.
Provide the ratio, exponent, or limiting value required.
See the verdict, condition, explanation, and sample terms.
When the absolute value of the common ratio is below 1, each term shrinks toward zero and the partial sums approach a finite number.
A convergent infinite series approaches a finite total as more terms are added. A divergent series does not approach a finite total; it may grow without bound, oscillate, or fail a required test.
The ratio test is conclusive only when L < 1 or L > 1. When L = 1, the test is inconclusive and another method must be used.
When does a geometric series converge?
An infinite geometric series converges when the absolute value of its common ratio is less than 1. Its sum is a/(1-r).
When does a p-series converge?
The series Σ1/nᵖ converges for p > 1 and diverges for p ≤ 1.
What does the ratio test tell me?
If L < 1, the series converges absolutely. If L > 1 or is infinite, it diverges. If L = 1, the result is inconclusive.
Does aₙ → 0 prove convergence?
No. It is necessary but not sufficient. If aₙ does not approach zero, the series diverges; if it does approach zero, another test is still needed.
Results are estimates generated locally in your browser using standard mathematical formulas. They are for informational and educational purposes only and are not a substitute for verification in coursework, examinations or professional contexts requiring certified accuracy. VisionVix accepts no liability for decisions based on this tool's output.
This calculator covers common textbook convergence rules and simplified test inputs. It does not replace a full symbolic analysis of an arbitrary sequence or series. Verify important coursework with your instructor or a trusted mathematics reference.
