Infinite Series Sum Calculator

The Infinite Series Sum Calculator finds the exact closed-form sum of eight classic infinite series — geometric, p-series, telescoping, alternating harmonic, Leibniz, Basel, exponential (eˣ), and alternating p-series — along with the convergence reason, the formula used, and a partial-sums table showing how quickly (or slowly) the series actually approaches that sum.

How to use it

1. Pick a series family from the dropdown.
2. Fill in that family's parameters — a first term and ratio for a geometric series, an exponent for a p-series, a gap for a telescoping series, and so on. Some families (alternating harmonic, Leibniz, Basel) have no parameters at all.
3. Choose how many partial sums to display.
4. Read the exact sum and formula, or the full step-by-step derivation for the reasoning behind the convergence verdict.

Why the general p-series and alternating p-series aren't naive partial sums

For most exponents p, Σ 1/nᵖ and Σ (-1)ⁿ⁻¹/nᵖ have no elementary closed form — their true "closed form" is the Riemann zeta function ζ(p) and the Dirichlet eta function η(p). A calculator that just adds up thousands of raw terms converges painfully slowly for p near 1 (the alternating harmonic series needs billions of terms for single-digit precision) and is flatly wrong for a divergent p-series. This tool instead:

• Evaluates ζ(p) with the Euler-Maclaurin summation formula — the same technique used by mpmath and PARI/GP — accurate to roughly 14 significant digits for any p > 1, with exact known values (π²/6, π⁴/90, π⁶/945, π⁸/9450, π¹⁰/93555) shown directly for even integer exponents.
• Evaluates η(p) by applying the Euler transform to the alternating series, which accelerates convergence from thousands of raw terms down to about 30 — accurate to roughly 10 significant digits for any p > 0.

The eight series

• Geometric — Σ a·rⁿ. Converges iff |r| < 1, summing to a·rⁿ⁰/(1 − r).
• p-series — Σ 1/nᵖ. Converges iff p > 1, summing to ζ(p) (the Riemann zeta function).
• Telescoping — Σ 1/(n(n + k)). Always converges, since 1/(n(n+k)) = (1/k)[1/n − 1/(n+k)] cancels every interior term.
• Alternating harmonic — Σ (-1)ⁿ⁻¹/n. Sums to ln 2.
• Leibniz — Σ (-1)ⁿ/(2n + 1). Sums to π/4 — one of the slowest-converging series in common use (arctan(1)).
• Basel problem — Σ 1/n². Sums to π²/6, Euler's famous 1734 result.
• Exponential — Σ xⁿ/n!. Converges for every real x, summing to eˣ.
• Alternating p-series — Σ (-1)ⁿ⁻¹/nᵖ. Converges (conditionally, for 0 < p ≤ 1) iff p > 0, summing to η(p) (the Dirichlet eta function; η(1) = ln 2 exactly).

To go the other direction — expand a function into a power series rather than sum one — see the Taylor Series Calculator, or check whether a series you already have converges with the Series Convergence Test Calculator.

Privacy

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