Test infinite series for convergence using ratio, root, comparison, integral, and alternating series tests with partial sum plots.
Enter the general term of an infinite series - something like 1/n^2 or (-1)^n/n - and this tool returns a verdict: converges, diverges, or inconclusive, with the test that settled it. It runs the standard battery - ratio, root, comparison, integral, p-series, and alternating series tests - then plots partial sums so the answer carries visual evidence. All computation happens in the browser; nothing is uploaded. The surprise most users hit: a divergent series like 1/n grows its partial sums so slowly that the plot looks flat, so a level tail is not proof of convergence.
A concrete input and expected output from the current implementation.
Input
1/n^2
Expected output
Converges (p-series test, p = 2 > 1). Partial sums: S_10 ~ 1.550, S_100 ~ 1.635, S_1000 ~ 1.644, approaching 1.6449, which equals pi^2/6.
1/n^2 is a p-series with p = 2 > 1, so it converges; its exact sum is pi^2/6 ~ 1.644934, and the partial sums approach it from below.
The ratio test takes the limit of |a_(n+1)/a_n|, the root test takes the limit of the nth root of |a_n|, the alternating test checks that terms decrease to zero while signs flip, and the integral test compares the series to the integral of its own term. A limit of exactly 1 is inconclusive for that test, so the tool falls through and reports the first test that yields a definite answer.
The tool accumulates terms and plots each partial sum S_N against N. A convergent series produces a plot that visibly levels off at a value the tool reports as the approximate sum; a divergent series keeps climbing, however slowly. The harmonic series 1/n, for instance, reaches only about 7.49 by S_1000, so the plot can look convergent while the series diverges.
Because the limit of |a_(n+1)/a_n| is exactly 1, and the ratio test is silent at that boundary: it cannot distinguish convergence from divergence there. The tool then falls through to another test - for 1/n^2, the p-series or integral test - which settles it. Inconclusive describes the test, not the series.
Almost never. Exact sums exist only for special families: geometric series and recognizable constants like pi^2/6 or Apery's constant. Everything else gets a computed partial sum, accurate to the displayed precision but possibly differing from the true limit in its last digits if the series converges slowly.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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