Infinite Series Calculator

An infinite series is the sum of infinitely many terms: Σ aₙ = a₁ + a₂ + a₃ + ... . The sum of an infinite series is defined as the limit of its partial sums. If this limit exists and is finite, the series converges; otherwise, it diverges.

A convergent series has a finite sum as the number of terms approaches infinity. For example, the geometric series with r=1/2 converges to 2. A divergent series either grows without bound or fails to approach a fixed value. The harmonic series Σ 1/n is a classic example that diverges (its partial sums grow to infinity, though very slowly).

A geometric series Σ arⁿ = a + ar + ar² + ... converges to S = a/(1 - r) when |r| < 1. When |r| ≥ 1, the series diverges. Examples: Σ (1/2)ⁿ = 1/(1-1/2) = 2; Σ 2ⁿ diverges because r = 2 > 1.

The p-series Σ 1/nᵖ converges when p > 1 and diverges when p ≤ 1. This is determined by the integral test. Common examples: Σ 1/n² converges (p=2), the harmonic series Σ 1/n diverges (p=1), Σ 1/√n diverges (p=1/2).

The alternating series test applies to series of the form Σ (-1)ⁿ aₙ or Σ (-1)ⁿ⁺¹ aₙ with aₙ > 0. The series converges if: (1) aₙ is decreasing (aₙ₊₁ ≤ aₙ) and (2) lim aₙ = 0 as n → ∞. The alternating harmonic series Σ (-1)ⁿ/n converges by this test, even though the regular harmonic series diverges.

The ratio test computes L = lim |aₙ₊₁ / aₙ|. If L < 1, the series converges absolutely; if L > 1, the series diverges; if L = 1, the test is inconclusive. This test is particularly useful for series involving factorials (n!) or exponentials (rⁿ).

A power series Σ aₙ xⁿ converges for |x| < R and diverges for |x| > R, where R is the radius of convergence. The radius can be found using the ratio test: R = 1 / lim sup |aₙ|^(1/n). At the endpoints |x| = R, convergence must be tested separately using other methods.

For alternating series that satisfy the convergence conditions, the error after truncating at N terms is bounded by the first omitted term: |Rₙ| ≤ aₙ₊₁. This makes alternating series very useful for numerical approximation because you know exactly how accurate your partial sum is.

Yes! A series converges conditionally if it converges but does not converge absolutely (the series of absolute values diverges). The alternating harmonic series Σ (-1)ⁿ/n is conditionally convergent — it converges to ln(2), but Σ 1/n diverges. The Riemann rearrangement theorem states that conditionally convergent series can be rearranged to converge to any real number!

The alternating harmonic series Σ (-1)ⁿ⁺¹ / n = 1 - 1/2 + 1/3 - 1/4 + ... converges to ln(2) ≈ 0.693147. This is a classic example of a conditionally convergent series with a beautiful result connected to natural logarithms.