Series Convergence Test Calculator

The Series Convergence Test Calculator tells you whether an infinite series converges absolutely, converges conditionally, or diverges — for a p-series, a geometric series, an alternating series, and three other standard series families. It shows the exact ratio-test and root-test limits, the divergence (nth-term) test, which rule actually decided the verdict, and the running partial sums.

How to use it

1. Pick a series family — p-series, geometric, alternating p-series, polynomial × geometric, exponential over factorial, or an alternating rational series.
2. Fill in that family's parameter(s) — an exponent, a ratio, or a base — and a starting index n₀.
3. Choose how many partial sums to show.
4. Read the Verdict and Decisive rule for the answer and the exact theorem behind it, or the Step-by-step derivation for the full picture.

Why not just sample the ratio test at a large n?

Most convergence calculators approximate the ratio and root tests by evaluating the series at one large n and hoping the trend has settled. That breaks down on the single most common convergence question there is: for a p-series (Σ 1/nᵖ), the ratio test limit and the root test limit are both exactly 1 for every value of p — so no amount of numeric sampling can ever tell a convergent p = 2 series apart from a divergent p = 1 (harmonic) series. This calculator classifies each family with its exact closed-form criterion instead — the p-series test, the geometric series test, the Alternating Series Test, or a limit comparison — and reports the exact ratio/root-test limit (0, |r|, or 1), not a floating-point guess at it.

The six series families

• p-series — Σ 1/nᵖ. Converges (absolutely) iff p > 1; diverges otherwise. This is the harmonic series at p = 1.
• Geometric series — Σ rⁿ. Converges (absolutely) iff |r| < 1.
• Alternating p-series — Σ (-1)ⁿ/nᵖ. Converges absolutely for p > 1, converges conditionally for 0 < p ≤ 1 (by the Alternating Series Test, even though the p-series test says Σ 1/nᵖ itself diverges there), and diverges for p ≤ 0.
• Polynomial × geometric — Σ nᵃ·rⁿ. The ratio test alone decides it whenever |r| ≠ 1 (the geometric factor always wins over the polynomial one); at |r| = 1 it reduces to a plain or alternating p-series in nᵃ.
• Exponential over factorial — Σ bⁿ/n!. Converges absolutely for every real b, because n! eventually outgrows any fixed exponential bⁿ — the ratio test limit is always 0. (This series sums to eᵇ.)
• Alternating rational — Σ (-1)ⁿ·n/(n²+a). Converges conditionally for every a ≥ 0: the terms shrink to 0, but the un-signed series diverges like the harmonic series by limit comparison.

What "absolute," "conditional," and "divergent" mean

• Converges absolutely — Σ |aₙ| also converges. The series converges regardless of the order its terms are added in.
• Converges conditionally — Σ aₙ converges, but Σ |aₙ| does not. Reordering the terms of a conditionally convergent series can change its sum (or make it diverge) — the Riemann series theorem.
• Diverges — the partial sums don't settle on a finite value.

Privacy

This calculator runs entirely in your browser. Your chosen series family, parameters, and starting index are never uploaded, logged, or stored.