Infinite Series Calculator

An infinite series is the sum of infinitely many terms of a sequence, written as Σaₙ from n=n₀ to ∞. A series is said to converge if the partial sums approach a finite limit, and diverge if they grow without bound or oscillate indefinitely.

A geometric series has the form Σ a·rⁿ, where r is the common ratio. It converges to a/(1−r) when |r| < 1, and diverges when |r| ≥ 1. The ratio r tells you how each term relates to the previous one.

A p-series has the form Σ 1/nᵖ. It converges when p > 1 and diverges when p ≤ 1. The classic harmonic series (p = 1) is a well-known example of a divergent p-series despite its terms approaching zero.

Sigma notation provides a compact, unambiguous way to express the sum of many terms. The expression Σ f(n) from n=a to ∞ specifies the formula f(n), the starting index a, and signals an infinite sum — making it far clearer than listing terms with ellipses.

A partial sum Sₙ adds only the first N terms of the series, providing a finite approximation. The infinite sum S is the limit of Sₙ as N → ∞. For convergent series, Sₙ gets arbitrarily close to S as N increases; the difference |S − Sₙ| is the truncation error.

A telescoping series is one where consecutive terms cancel, leaving only a finite number of terms. The series Σ 1/n(n+1) = Σ (1/n − 1/(n+1)) collapses to 1 − 1/(N+1), which converges to 1 as N → ∞.

The alternating series test (Leibniz criterion) states that a series of the form Σ (−1)ⁿ bₙ converges if the terms bₙ are positive, decreasing, and approach zero. The error of a partial sum is bounded by the first omitted term, making it easy to estimate accuracy.

Common errors include assuming that if terms go to zero the series must converge (the harmonic series disproves this), confusing the common ratio r with the first term a in geometric series, and forgetting to check that |r| < 1 before applying the geometric sum formula S = a/(1−r).