What is summation (sigma notation)?

Summation is the addition of a sequence of numbers, written with the Greek letter sigma: Σ. For example, Σᵢ₌₁ⁿ i = 1+2+3+...+n = n(n+1)/2. The notation specifies the variable, starting value, ending value, and the expression to sum.

What is the formula for the sum of first n numbers?

Σᵢ₌₁ⁿ i = n(n+1)/2. For example, 1+2+3+...+100 = 100×101/2 = 5050. This formula was famously derived by young Carl Friedrich Gauss.

What is the sum of squares formula?

Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6. For n=10: 1²+2²+...+10² = 10×11×21/6 = 385.

What is the sum of cubes formula?

Σᵢ₌₁ⁿ i³ = [n(n+1)/2]². Remarkably, the sum of cubes equals the square of the sum of first n numbers. For n=5: 1³+2³+...+5³ = 225 = 15².

What is the sum of a geometric series?

Σᵢ₌₀ⁿ arⁱ = a(1−rⁿ⁺¹)/(1−r) for r≠1. If |r|<1 and n→∞, the infinite sum is a/(1−r). For a=1, r=1/2: 1+1/2+1/4+...→2.

What is a partial sum?

A partial sum Sₙ is the sum of the first n terms. S₁=a₁, S₂=a₁+a₂, S₃=a₁+a₂+a₃, etc. Plotting partial sums reveals whether a series converges or diverges.

What is the harmonic series?

The harmonic series is Σ 1/i = 1 + 1/2 + 1/3 + 1/4 + ... It diverges but extremely slowly. The partial sum Hₙ ≈ ln(n) + γ where γ ≈ 0.5772 is the Euler-Mascheroni constant.

How do you evaluate a custom summation?

Enter a formula using i as the variable, like i^2+3*i or 1/i^2 or (-1)^i/i. The calculator evaluates each term from the lower to upper bound and sums them.

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Summation: The Complete Guide

What Is Summation?

Summation is the process of adding a sequence of numbers together, expressed with the Greek capital letter sigma (Σ). The notation Σᵢ₌ₘⁿ f(i) means "evaluate f(i) for each integer i from m to n, then add all the results." Summation is one of the most fundamental operations in mathematics — it connects arithmetic to calculus, underpins statistics (mean, variance), and powers algorithms in computer science.

Σᵢ₌₁ⁿ i = n(n+1)/2 — Gauss's formula
Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
Σᵢ₌₁ⁿ i³ = [n(n+1)/2]² = (Σ i)²
Σᵢ₌₀ⁿ rⁱ = (1−rⁿ⁺¹)/(1−r), r ≠ 1

Gauss's Formula: Sum of First n Numbers

Young Carl Friedrich Gauss astonished his teacher by instantly computing 1+2+3+…+100 = 5050. His insight: pair the first and last terms (1+100=101), second and second-to-last (2+99=101), forming 50 pairs of 101, totalling 5050. The general formula is Σᵢ₌₁ⁿ i = n(n+1)/2. This arithmetic series is the foundation for understanding all other power sums.

Example: Sum of 1 to 100

Σᵢ₌₁¹⁰⁰ i = 100 × 101 / 2 = 5,050. Average term = 50.5.

Sum of Squares and Cubes

The sum of squares Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6 appears in statistics (variance), physics (moments of inertia), and algorithm analysis. The sum of cubes has the elegant identity Σᵢ₌₁ⁿ i³ = [n(n+1)/2]² — the sum of cubes equals the square of the sum of natural numbers. This is Nicomachus's theorem. For n=5: 1³+2³+3³+4³+5³ = 225 = 15² = (1+2+3+4+5)².

Geometric Series

A geometric series has a constant ratio r between consecutive terms: Σᵢ₌₀ⁿ arⁱ = a(1−rⁿ⁺¹)/(1−r). When |r| < 1 and n → ∞, the infinite sum converges to a/(1−r). Geometric series model compound interest, radioactive decay, fractal geometry, and digital signal processing. For example, 1 + 1/2 + 1/4 + 1/8 + … = 2.

Harmonic and Alternating Series

The harmonic series Σ 1/i = 1 + 1/2 + 1/3 + … diverges but extremely slowly: Hₙ ≈ ln(n) + γ, where γ ≈ 0.5772 is the Euler-Mascheroni constant. The alternating harmonic Σ(−1)ⁱ⁺¹/i = 1 − 1/2 + 1/3 − 1/4 + … converges to ln(2) ≈ 0.6931. Other famous alternating sums: Σ(−1)ⁱ/(2i+1) = π/4 (Leibniz formula).

Partial Sums and Convergence

A partial sum Sₙ is the sum of the first n terms. Plotting partial sums reveals convergence behavior: if Sₙ approaches a finite limit, the series converges; if Sₙ grows without bound, it diverges. The partial sum sequence is the bridge between finite sums and infinite series — it's how mathematicians make sense of adding infinitely many numbers.

Pro Tip

For custom mode, enter any expression using i. Examples: "i^2 + 3*i" (polynomial), "1/i^2" (Basel-like), "(-1)^i / (2*i+1)" (Leibniz), "2^i" (exponential), "sqrt(i)". Use parentheses for clarity.

Properties of Summation

Summation is linear: Σ(af+bg) = aΣf + bΣg. It's telescoping when consecutive terms cancel: Σ(f(i)−f(i−1)) = f(n)−f(0). Summation connects to integration via the Euler-Maclaurin formula, which bridges discrete sums with continuous integrals, giving highly accurate approximations for large sums.

Applications

Summation appears everywhere: Statistics (mean = Σxᵢ/n, variance = Σ(xᵢ−μ)²/n), Finance (present value of annuities, compound interest), Physics (work as Σ Fᵢ·Δxᵢ, electric potential), Computer science (algorithm analysis, Big-O via summation), and Signal processing (discrete Fourier transform). Understanding summation is a prerequisite for calculus, where integration replaces discrete sums with continuous ones.

How to Use This Calculator

Select a series type from the 8 options. Set the lower and upper bounds. For Geometric series, enter the first term (a) and ratio (r). For Power mode, enter the exponent p. For Custom, type any formula using i. Click "Calculate Sum" to see the total, sigma notation, bar chart (blue bars = terms, green line = partial sums), closed-form formula, and step-by-step breakdown.

Watch Out

The harmonic series diverges — it grows beyond any bound. Very large upper bounds (n > 10,000) may slow the calculator. For huge sums, use the closed-form formulas.

Summation Calculator

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