Understanding the Test
The test for divergence is a first check for infinite series. It studies the limit of the general term. A series cannot converge unless its terms approach zero. When the limit is nonzero, infinite, or missing, the series diverges at once. When the limit is zero, this test stops. Another method is then needed.
Why the Limit Matters
Partial sums can settle only when added terms become very small. If terms keep a fixed size, the sums keep moving. If terms grow, the sums move faster. If terms alternate without shrinking, the sums do not settle. The divergence test captures this basic requirement before harder work begins.
What This Calculator Does
This calculator accepts common term structures. You can check power terms, rational powers, geometric terms, alternating powers, and factorial comparisons. It reports the estimated limit form. It also explains why the conclusion follows. The output is useful for homework checks, lesson examples, and quick review.
Reading the Result
A result marked divergent is final for this test. It means the nth term limit failed to become zero. A result marked inconclusive is not a convergence claim. It only means the term limit reached zero. You may still need the p test, ratio test, root test, comparison test, or integral test.
Best Study Practice
Write the general term before using any tool. Identify the dominant part of the expression. For rational powers, compare numerator and denominator degrees. For geometric terms, inspect the ratio size. For alternating terms, check whether the magnitude shrinks. Then compare your reasoning with the calculator steps.
Avoiding Common Errors
Many students say a zero limit proves convergence. That is false. The harmonic series has terms that go to zero, but it diverges. Others forget that a missing limit also proves divergence. Always separate the term test from stronger tests. Use it as a gatekeeper, not as the final judge for every series.
Final Note
The tool does not replace full proof writing. It supports the first decision. Always record the limit statement, the test name, and the conclusion. Clear notes help teachers see your method. They also make later tests easier when this first test gives an inconclusive answer for each problem carefully.