Summation Calculator
summation calculator. Get instant, accurate results. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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Formula: Σ from i=start to end of f(i) = f(start) + f(start+1) + ... + f(end)
Worked example — 338,350
Σ from i=start to end of f(i) = f(start) + f(start+1) + ... + f(end)
Sigma notation compactly represents the sum of a series. The expression is evaluated for each integer value of i from start to end. Common formulas: Σi = n(n+1)/2, Σi² = n(n+1)(2n+1)/6.
Problem:Σ i² from i=1 to 100
Solution:Formula: 100×101×201/6 = 338,350
Result:338,350
Σi (1 to n) = n(n+1)/2. Σi² = n(n+1)(2n+1)/6. Σi³ = [n(n+1)/2]². Σaʳ = a(rⁿ-1)/(r-1) for geometric series.
A finite summation (Σ from start to end) always produces a definite number when the expression is defined at each integer index. An infinite series (Σ from n=1 to ∞) only converges to a finite value if the terms decrease fast enough. For example, Σ 1/2ⁿ = 1 (converges), but Σ 1/n diverges. Use the Series Convergence calculator to test whether an infinite series converges.
In statistics, sigma notation appears in mean (x̄ = Σxᵢ/n), variance (σ² = Σ(xᵢ - x̄)²/n), and standard deviation calculations. In probability, expected value is E[X] = Σ x·P(x) summed over all outcomes. Summation notation compactly expresses operations over all data points or outcomes, making it foundational for statistical formulas involving large datasets.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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