Series Convergence Calculator (Converge or Diverge)

Does the sum settle, or run away?

An infinite series adds up infinitely many terms. The whole question is whether that running total approaches a finite number (it converges) or grows without bound or oscillates forever (it diverges). This calculator checks the three families that show up most in a first calculus course, and reports the verdict, the test that settles it, and the sum when one exists.

Geometric series

Σ a·rⁿ converges when |r| < 1, and then the sum is exactly a / (1 − r). At |r| ≥ 1 it diverges. This is the one series with a clean closed-form sum, which is why 1 + ½ + ¼ + ⅛ + … lands precisely on 2.

p-series

Σ 1/nᵖ converges when p > 1 and diverges when p ≤ 1. The boundary case p = 1 is the harmonic series, which diverges even though its terms shrink to zero, a fact Nicole Oresme proved back in the 14th century. At p = 2 the sum is the famous π²/6, the Basel problem that Euler cracked in 1735.

Alternating series

Σ (−1)ⁿ⁺¹/nᵖ converges for any p > 0, because the terms flip sign and shrink toward zero. The alternating harmonic series (p = 1) converges to ln 2 ≈ 0.6931, even though dropping the signs turns it into the divergent harmonic series. That gap between ordinary and absolute convergence is one of the stranger facts in the subject.

The ratio test, in one line

For nastier terms built from factorials or exponentials, compute L = lim |aₙ₊₁ / aₙ|. If L < 1 the series converges, if L > 1 it diverges, and if L = 1 the test tells you nothing and you reach for another tool. The ratio test is usually the first thing to try when a term has n! or a constant raised to the n.