Series convergence checker
Apply supported convergence tests after checking the original term domain at every included integer index. The report distinguishes a proved verdict from an inconclusive result; supported geometric series also get an exact sum.
Terms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests. Use n as the index. Add a start such as n=2: before the term if needed. Up to 1,000 characters. Review the settings before running.
Your result
How to use Series convergence checker
- Type the general term, such as 3/2^n, or set the start with a prefix such as n=2: 1/(n*ln(n)^2).
- Run the checker to apply the tests in order.
- Read the verdict, the deciding test and the preconditions of every test.
- Download the JSON report or the CSV files of tests and partial sums.
Useful next steps
Terms are limited to 1,000 characters and one index variable. An explicit prefix such as n=2: uses a starting index from 0 to 9999 and is never moved. Without a prefix, the displayed start may be moved past rational poles or to 2 for the supported logarithmic pattern. Full integer-domain proofs support rational conditions and selected integer-power patterns; some valid expressions are refused when their domain cannot be proved. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of partial sums and a CSV of tests.
Tests used
The checker applies the nth-term test, the geometric series test, the ratio and root tests, limit comparison with a p-series using the exact leading term c*n^p, direct comparison for bounded factors such as sin(n), the alternating series test with exact checks that the terms decrease to zero, and the integral test for 1/(n*ln(n)^p). Method: OpenStax, Calculus Volume 2, sections 5.2 to 5.6 (2016).
Preconditions
Original denominators and other supported domain conditions are checked on the entire included integer tail before cancellation or zero-factor simplification. An included undefined term is rejected even when it occurs beyond the partial-sum table. When the full domain cannot be proved by the supported method, the checker returns a recoverable error. A test decides a verdict only when its preconditions are proved; otherwise the verdict remains inconclusive.
Partial sums
Partial sums are listed for orientation. They are exact where the term evaluator returns exact rational values and otherwise use decimals, even when the terms are mathematically rational. They never prove convergence or divergence.