Summation Calculator

How to Use This Calculator

1. Set the lower bound a and upper bound b for your summation index.
2. Choose f(i) from the dropdown (i, i², i³, constant, or odd numbers).
3. The result shows Σ f(i), term count, and closed-form value instantly.
4. Use Common Forms tab to compare all standard formulas for the same n.
5. Use Telescoping tab to verify cancellation in Σ 1/(i(i+1)).
6. The Professional tab handles harmonic and Basel series with convergence analysis.

Formula

Example

Frequently Asked Questions

• What does the summation symbol mean and how do I read Σ notation? ▼
  The uppercase Greek letter Σ (sigma) is the universal symbol for summation, meaning "add up all the following terms." Sigma notation is written as Σᵢ₌ₐᵇ f(i), read as "the sum of f(i) from i equals a to b." The variable i is the index, a is the lower bound (starting value), b is the upper bound (ending value), and f(i) is the expression evaluated at each integer i. For example, Σᵢ₌₁⁵ i means 1+2+3+4+5 = 15. The index i is a dummy variable — you could use j, k, or n and get the same result. Sigma notation is a compact way to express sums that would otherwise require writing out many terms, and it is fundamental in calculus, statistics, discrete mathematics, and computer science. Understanding sigma notation unlocks power series, probability distributions, and algorithm analysis. The summation index always takes integer values, stepping by 1 from the lower to upper bound unless otherwise stated.
• How do I evaluate sums of common sequences like 1+2+...+n? ▼
  Many common summations have elegant closed-form formulas. The most famous is the sum of the first n integers: Σᵢ₌₁ⁿ i = n(n+1)/2. For n=100 this gives 5050, famously computed by Gauss as a child. The sum of squares: Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6. For n=10 this equals 385. The sum of cubes: Σᵢ₌₁ⁿ i³ = [n(n+1)/2]², which equals the square of the triangular number — a beautiful identity meaning the cube sum always equals a perfect square. The sum of the first n odd numbers, Σᵢ₌₁ⁿ (2i−1), equals n², which is why perfect squares are sums of consecutive odd numbers (1, 1+3=4, 1+3+5=9, etc.). For arbitrary bounds a to b, subtract: Σᵢ₌ₐᵇ f(i) = Σᵢ₌₁ᵇ f(i) − Σᵢ₌₁ᵃ⁻¹ f(i). These closed forms allow instant evaluation without iterating.
• What is the difference between finite and infinite summation? ▼
  A finite sum has a specific upper bound b, producing a definite numerical answer. An infinite sum (infinite series) extends forever: Σᵢ₌₁^∞ f(i). Whether an infinite series has a finite answer depends on how fast terms shrink. If terms decrease fast enough, the partial sums approach a limit and we say the series converges. For example, Σ(1/2ⁿ) = 1+1/2+1/4+... converges to 2. But Σ(1/n), the harmonic series, diverges to infinity despite terms approaching zero — a counterintuitive result first proved by Oresme around 1350. The key distinction: a finite sum always has an answer; an infinite sum requires convergence analysis before evaluation. Partial sums S_n = Σᵢ₌₁ⁿ f(i) serve as finite approximations of infinite series. Zeno's paradox highlights why infinite sums fascinated ancient thinkers: infinitely many positive steps can yield a finite total distance.
• When does an infinite sum converge vs diverge? ▼
  Convergence tests determine whether an infinite series has a finite sum. The divergence test is the first check: if terms do not approach zero (lim f(i) ≠ 0), the series diverges. However, terms approaching zero is necessary but not sufficient — the harmonic series Σ(1/n) diverges despite 1/n→0. Key tests: the ratio test (if |a_{n+1}/aₙ| < 1, converges — useful for factorials and exponentials); the integral test (series converges iff the corresponding improper integral converges); the comparison test (dominate or bound by a known series); and the p-series test (Σ1/nᵖ converges if p>1, diverges if p≤1). The Basel problem (Σ1/n² = π²/6) converges since p=2>1, yielding Euler's remarkable 1734 result. Geometric series Σrⁿ converges to 1/(1−r) when |r|<1. Alternating series with decreasing terms converge by the alternating series test.
• How is summation used in calculus and statistics? ▼
  Summation is the discrete foundation underlying continuous calculus. Riemann sums — Σ f(xᵢ)Δx over many subintervals — approximate definite integrals, and as Δx→0 the sum becomes the integral. The integral sign ∫ deliberately resembles an elongated S for "sum." In statistics, summation is omnipresent: the mean is (Σxᵢ)/n, variance is Σ(xᵢ−μ)²/n, and covariance is Σ(xᵢ−x̄)(yᵢ−ȳ)/(n−1). Least-squares regression minimizes Σ(yᵢ−ŷᵢ)². In probability, expected value is Σ x·P(x) over all discrete outcomes. Taylor series express functions as infinite sums of polynomial terms: eˣ = Σ xⁿ/n!, sin(x) = Σ (−1)ⁿx^(2n+1)/(2n+1)!. In computer science, algorithm complexity analysis requires evaluating summations — Σᵢ₌₁ⁿ i = O(n²) explains quadratic behavior of nested loops. Fourier series decompose periodic signals into sums of sine/cosine waves.

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