Summation Calculator (Σ)
Evaluate summation expressions with custom formulas, bounds, and step values.
Compute Σ f(n) from n=start to n=end. Use n as the variable.
What Is the Summation Calculator (Σ)?
The Summation Calculator evaluates the sum Σf(n) for any formula f(n) over a specified range of n values. Enter any expression in n — polynomials, powers, fractions — and the calculator evaluates it term by term, displaying each term and the running total.
Formula
How to Use
Type a formula using n as the variable (e.g., n^2+1, 1/n, 2^n). Set the start and end values for n. Click Calculate to see the full expansion with each term and the final sum. Use the example buttons for common formulas like n, n², n³.
Example Calculation
Sum of squares from 1 to 5: Σn² (n=1 to 5) n=1: 1 n=2: 4 n=3: 9 n=4: 16 n=5: 25 Total = 55 Verify: 5×6×11/6 = 55 ✓ Geometric sum: Σ2^n (n=0 to 4) = 1+2+4+8+16 = 31
Understanding Summation (Σ)
Summation notation (Σ, the capital Greek letter sigma) is the compact way to write sums with many terms. It is fundamental to calculus (Riemann sums approximate integrals), statistics (mean = Σxᵢ/n), discrete mathematics, and algorithm analysis.
Closed-form expressions for sums — like Σk = n(n+1)/2 — are elegant mathematical discoveries. Gauss reportedly computed the sum 1+2+...+100 as a child by pairing terms: (1+100) + (2+99) + ... + (50+51) = 50 × 101 = 5050.
Summation is the discrete analogue of integration. As the step size approaches zero, a Riemann sum converges to a definite integral. This connection between sums and integrals is one of the deep insights of calculus.
Frequently Asked Questions
What notation does the formula accept?
Use standard math notation: n^2 for n², n^3 for n³, 1/n for fractions, 2^n for 2ⁿ, n*(n+1) for products. The calculator evaluates JavaScript-compatible expressions.
What is the maximum range I can sum over?
For performance, the tool evaluates up to a few thousand terms directly. For very large ranges, use closed-form formulas (like n(n+1)/2 for Σn) instead of term-by-term summation.
What is a harmonic series?
The harmonic series Σ1/n (from n=1 to ∞) diverges — it grows without bound, just very slowly. Its partial sum is approximately ln(n) + 0.5772 (Euler-Mascheroni constant).
What is a geometric series?
A geometric series has a constant ratio between consecutive terms: a + ar + ar² + ... + arⁿ. If |r| < 1, the infinite sum converges to a/(1−r).