About Series Divergence Testing
A series can fail before any detailed convergence test begins. The nth term divergence test checks the sequence of terms. It asks one direct question. Does the term approach zero as n grows without bound? If the answer is no, the infinite sum diverges. If the answer is yes, the result is only undecided.
Why the Limit Matters
An infinite series adds terms one after another. For a stable total, later terms must become very small. They must approach zero. A nonzero tail keeps adding weight forever. A term with no limit also prevents a settled sum. This rule is necessary, not sufficient. Many zero term series still diverge.
What This Tool Does
This calculator evaluates common term models used in statistics, probability, and mathematical analysis. It supports rational powers, geometric terms, logarithmic ratios, decay with an offset, and alternating terms. It estimates sample values and compares them with the found limit. The result states whether divergence is proven or whether another test is required.
Interpreting the Answer
A divergent result is final for the chosen model. It means the term limit is nonzero, infinite, or not defined. An inconclusive result needs more work. Use comparison, ratio, root, integral, or alternating tests next. Never claim convergence from this test alone.
Practical Use in Statistics
Series appear in estimators, probability tails, likelihood expansions, and simulation errors. A quick divergence screen can prevent wasted analysis. It also helps students see why a vanishing term is only a starting requirement. Use exact formulas where possible. Use numeric samples as support, not proof.
Good Modeling Habits
Choose the model that matches your term. Enter signs carefully. Use enough decimal precision for close limits. Increase sample size when terms change slowly. Check the formula section before relying on the verdict. The test is simple, yet it protects many advanced calculations from a wrong first step.
Limits of the Test
The method never measures the whole sum. It only examines the individual term. A zero limit can hide harmonic behavior, slow logarithmic decay, or other divergent patterns. Treat the output as a gatekeeper. It rules out impossible convergence. It does not certify safe convergence. Pair it with a second test when needed.