Infinite Sum Calculator

An infinite sum, or infinite series, is the addition of infinitely many terms in a sequence. Mathematically, it is the limit of the partial sums. If the partial sums approach a specific value as you add more terms, the series is said to converge to that sum.

Convergence depends on the behavior of the terms as n approaches infinity. Our calculator automatically applies several tests, including the Ratio Test (looking at the ratio of consecutive terms), the Root Test, and the p-Series test. Generally, if the terms do not approach zero, the series diverges.

For a geometric series with first term 'a' and common ratio 'r', the sum is S = a / (1 - r), provided that the absolute value of the ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and has no finite sum.

The Ratio Test calculates the limit L = lim |aₙ₊₁ / aₙ| as n → ∞. If L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive and another method, like the Integral Test or Comparison Test, must be used.

A sequence is a list of numbers in a specific order (e.g., 1, 1/2, 1/4...). A series is the sum of the terms in a sequence (e.g., 1 + 1/2 + 1/4...). Our calculator analyzes the sequence terms to find the total sum of the series.

Yes, our advanced engine supports factorials (n!), powers, and standard trigonometric or logarithmic functions within the general term (aₙ). It uses numerical approximations for complex limits when exact symbolic forms aren't available.

A p-series is any series in the form Σ 1/nᵖ. It converges if and only if p > 1. For example, Σ 1/n² (the Basel problem) converges to π²/6, while Σ 1/n (the Harmonic series) diverges to infinity.