Series Convergence Calculator

Series Convergence: The Complete Guide

What Is an Infinite Series?

An infinite series is the sum of infinitely many terms: S = a₁ + a₂ + a₃ + ... = Σaₙ. The key question is whether this sum approaches a finite number (converges) or grows without bound (diverges). The partial sum Sₙ = a₁ + a₂ + ... + aₙ gives the sum of the first n terms. If the sequence of partial sums {S₁, S₂, S₃, ...} approaches a finite limit L, the series converges to L. Otherwise, it diverges.

Geometric Series

A geometric series has the form Σarⁿ where a is the first term and r is the common ratio. This is one of the few series with a simple closed-form sum. It converges if |r| < 1 with sum S = a/(1−r). It diverges if |r| ≥ 1. Geometric series appear in compound interest, probability, fractals, and signal processing. For example, the series 1 + 1/2 + 1/4 + 1/8 + ... (a=1, r=1/2) converges to 2.

p-Series and Harmonic Series

The p-series Σ(1/nᵖ) converges if and only if p > 1. The boundary case p = 1 gives the harmonic series Σ(1/n) = 1 + 1/2 + 1/3 + 1/4 + ..., which diverges — albeit very slowly. This is a famous result: even though the terms approach zero, the partial sums grow without bound (reaching 10 after about 12,367 terms and 20 after about 272 million terms). For p = 2, the series converges to the remarkable value π²/6 ≈ 1.6449 (the Basel problem, solved by Euler in 1735).

The Ratio and Root Tests

The ratio test examines L = lim|aₙ₊₁/aₙ|. If L < 1, the series converges absolutely; if L > 1, it diverges; if L = 1, the test is inconclusive. The root test computes L = lim(|aₙ|^(1/n)). Same decision criteria apply. The ratio test works especially well for series involving factorials, exponentials, and products, while the root test excels for series of the form (f(n))ⁿ. Neither test can resolve p-series (both give L = 1).

Alternating Series Test (Leibniz Test)

An alternating series has terms that alternate in sign: Σ(−1)ⁿbₙ. The alternating series test says it converges if (1) bₙ is eventually decreasing and (2) lim bₙ = 0. The alternating harmonic series Σ(−1)ⁿ⁺¹/n = 1 − 1/2 + 1/3 − 1/4 + ... converges to ln(2) ≈ 0.6931, even though the harmonic series itself diverges. The error after n terms is bounded by |aₙ₊₁|, giving a useful approximation guarantee.

Telescoping Series

A telescoping series has terms that cancel in pairs when written as partial fractions. The classic example Σ 1/(n(n+1)) = 1/(1·2) + 1/(2·3) + 1/(3·4) + ... telescopes because 1/(n(n+1)) = 1/n − 1/(n+1). The partial sum collapses to Sₙ = 1 − 1/(n+1), so the series converges to 1. Telescoping series are powerful because they yield exact sums, not just convergence verdicts.

Comparison and Integral Tests

The comparison test compares a series to a known benchmark: if 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges. If aₙ ≥ bₙ ≥ 0 and Σbₙ diverges, then Σaₙ diverges. The limit comparison test checks lim(aₙ/bₙ): if this limit is positive and finite, both series share the same convergence behavior. The integral test connects series to improper integrals: if f(n) = aₙ where f is positive, continuous, and decreasing, then Σaₙ and ∫f(x)dx both converge or both diverge.

Absolute vs Conditional Convergence

A series converges absolutely if Σ|aₙ| converges. It converges conditionally if Σaₙ converges but Σ|aₙ| diverges. Absolute convergence is stronger: any absolutely convergent series is convergent, but not vice versa. The alternating harmonic series converges conditionally (it converges, but the harmonic series diverges). Remarkably, the Riemann rearrangement theorem shows that a conditionally convergent series can be rearranged to converge to any desired value, or to diverge — a fascinating and counterintuitive result.

Series in Science and Engineering

Infinite series are foundational in applied mathematics. Taylor series represent functions as infinite polynomials: eˣ = Σxⁿ/n!, sin(x) = Σ(−1)ⁿx²ⁿ⁺¹/(2n+1)!. Fourier series decompose periodic signals into sine and cosine waves, essential in signal processing, acoustics, and image compression (JPEG uses the discrete cosine transform). In physics, perturbation theory uses series expansions to approximate solutions to complex equations. In finance, geometric series underpin present value calculations, annuity pricing, and perpetuity valuations.

How to Use This Calculator

Select a series type: Geometric (Σarⁿ), p-Series (Σ1/nᵖ), Alternating (Σ(−1)ⁿ/nᵖ), Telescoping (Σ1/(n(n+k))), Exponential (Σxⁿ/n!), or Power·Ratio (Σnᵃ·rⁿ). Enter the required parameters and click "Test Convergence." The calculator displays the verdict (converges/diverges), the exact or approximate sum, a partial sums graph showing the series behavior, step-by-step test application, and a partial sums table for the first 30 terms.