Sequence convergence explorer
Tabulate the terms of a sequence exactly and decide its limit, monotonicity and bounds where an exact method applies. Decimal tails are shown separately as evidence.
Terms are limited to 1,000 characters and one index variable; 1 to 500 terms are tabulated, exactly while values stay under 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms. Use n as the index, such as n/2^n or (1 + 1/n)^n. Up to 1,000 characters. Review the settings before running.
Your result
How to use Sequence convergence explorer
- Type the general term, such as (2n^2 + 1)/(n^2 - 3).
- Choose how many terms to tabulate.
- Run the explorer and read the limit, the method, the monotonicity and the bounds.
- Download the JSON report or the CSV of terms.
Useful next steps
Terms are limited to 1,000 characters and one index variable; 1 to 500 terms are tabulated, exactly while values stay under 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of terms.
Limits
Fractions of polynomials are decided by comparing degrees. Expressions with roots use the exact leading term c*n^p, including the conjugate trick for sqrt(n^2 + n) - n. Geometric factors such as n/2^n, the limit of (1 + a/n)^(b n) = exp(a b), n^(1/n) and bounded factors such as sin(n)/n are recognised. Method: OpenStax, Calculus Volume 2, section 5.1 Sequences (2016).
Monotone and bounded
For fractions of polynomials the sign of a(n+1) - a(n) is constant beyond its last real root, which is found exactly; the earlier terms are compared exactly to find where the pattern starts. Bounds then follow from the first terms and the limit.
Evidence versus proof
When no exact method applies the limit is reported as not determined. The decimal tail of the table never decides the answer.