Summation Calculator | Sigma Notation
Evaluate summation expressions with custom formulas, bounds, and step values.
Compute Σ f(n) for n = start to end. Use n as the variable, ^ for power.
Common series:
What Is the Summation Calculator | Sigma Notation?
The Summation Calculator evaluates Σ f(n) for any formula using n as the variable over any integer range. It computes the exact sum term-by-term, displays the first 10 terms and last 5 terms in a breakdown table, and draws a running partial-sums chart. When the formula matches a known closed form (Σn, Σn², Σn³, Σ2ⁿ), it identifies and displays the closed-form result for verification.
- ›Use n as the variable; ^ for powers; * for multiplication; 1/n for fractions
- ›Supported operators: +, −, *, /, ^ (power)
- ›Closed forms auto-detected for Σn, Σn², Σn³, Σ2ⁿ
- ›Range up to 50,000 terms; larger ranges use closed-form formulas
- ›Running sum chart shows convergence or divergence visually
Formula
Summation Formulas & Closed Forms
Σn
n(n+1)/2 (triangular numbers)
Σn²
n(n+1)(2n+1)/6
Σn³
[n(n+1)/2]² (square of Σn)
Σaⁿ
a(aⁿ−1)/(a−1) (geometric)
Σ1/n
≈ ln(n) + 0.5772 (harmonic)
Σ1/n²
→ π²/6 ≈ 1.6449 (Basel)
How to Use
- 1Enter a formula in the f(n) field using n as the variable (e.g., n^2, 1/n, n*(n+1))
- 2Set the start and end values of n (integers)
- 3Click Calculate Sum
- 4The total sum, term count, and individual terms are displayed
- 5Click a Common Series preset to instantly load Σn, Σn², Basel series, etc.
- 6The partial sums chart shows how the running total builds up across the range
Example Calculation
Sum of squares: Σn² from n=1 to 10
Closed form: 10×11×21/6 = 2310/6 = 385 ✓
Basel problem: Σ1/n² from n=1 to 1000
Euler's exact result: π²/6 ≈ 1.6449340668...
Error after 1000 terms: ~0.001
Gauss and the sum of 1 to 100
Young Carl Friedrich Gauss reportedly computed 1+2+...+100 = 5050 in seconds by pairing: (1+100) + (2+99) + ... + (50+51) = 50 pairs × 101 = 5050. This is exactly the closed form n(n+1)/2 = 100×101/2 = 5050, a formula that remains one of the most elegant in all of mathematics.
Understanding Summation | Sigma Notation
Common Summation Identities
| Series | Formula f(n) | Closed Form | Value (n=10) |
|---|---|---|---|
| Arithmetic | n | n(n+1)/2 | 55 |
| Sum of squares | n² | n(n+1)(2n+1)/6 | 385 |
| Sum of cubes | n³ | [n(n+1)/2]² | 3025 |
| Geometric (r=2) | 2ⁿ (n=0) | 2ⁿ⁺¹−1 | 1023 |
| Harmonic | 1/n | ≈ ln(n)+γ | 2.9290 |
| Basel (partial) | 1/n² | → π²/6 ≈ 1.6449 | 1.5498 |
| Alternating harm. | (−1)ⁿ/n | → −ln(2) ≈ −0.693 | −0.6456 |
| Consecutive prod. | n(n+1) | n(n+1)(n+2)/3 | 440 |
Frequently Asked Questions
What formula syntax does the calculator accept?
- ›n^2 → n²; n^3 → n³; n^0.5 → √n
- ›1/n → 1/n (harmonic terms)
- ›n*(n+1) → n(n+1) (consecutive product)
- ›2^n → 2ⁿ (geometric terms)
- ›((-1)^n)/n → alternating harmonic series
- ›Whitespace is ignored; operators follow standard maths precedence
What is a closed-form sum formula?
Closed-form formulas avoid rounding accumulation from summing millions of terms. They are exact for integer n and were known to ancient Greek and Indian mathematicians.
- ›Σk = n(n+1)/2: sum 1+2+...+n (Gauss's formula)
- ›Σk² = n(n+1)(2n+1)/6: sum of squares
- ›Σk³ = [n(n+1)/2]² = (Σk)², remarkable identity
- ›Geometric: Σrᵏ (k=0 to n) = (rⁿ⁺¹−1)/(r−1)
What is the harmonic series and does it converge?
- ›Harmonic partial sum H_n ≈ ln(n) + 0.5772 (Euler-Mascheroni constant γ)
- ›Grows extremely slowly: need ~10^43 terms to exceed 100
- ›Contrast: Σ1/n² converges to π²/6 ≈ 1.6449 (Basel problem, solved by Euler)
- ›Σ1/n^p converges for p > 1 (p-series convergence test)
What is a geometric series and when does it converge?
- ›Sum of 1 + 1/2 + 1/4 + ... = 1/(1−0.5) = 2 (converges)
- ›Sum of 1 + 2 + 4 + ... diverges (|r| = 2 ≥ 1)
- ›1 + r + r² + ... + rⁿ = (rⁿ⁺¹−1)/(r−1) for r ≠ 1
- ›Applications: compound interest, decay processes, repeating decimals
How is summation related to integration?
The correspondence between sums and integrals is one of calculus's deepest results. Many definite integrals can be evaluated by first computing a sum and taking the limit.
- ›Riemann sum: Σf(xᵢ)Δx → ∫f(x)dx as n → ∞
- ›Σ1/n (harmonic) ↔ ∫1/x dx = ln(x) (explains harmonic divergence)
- ›Euler-Maclaurin formula links sums and integrals precisely
- ›Discrete → continuous: difference equations ↔ differential equations
What is the Basel problem?
- ›Σ1/n² = π²/6 ≈ 1.6449340668... (Basel problem, Euler 1734)
- ›Σ1/n⁴ = π⁴/90; Σ1/n⁶ = π⁶/945 (all even powers of π)
- ›Odd powers (Σ1/n³ = ζ(3) ≈ 1.202) are not known to involve π
- ›Riemann zeta function ζ(s) generalises these sums to complex s
Is this summation calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser.