Summation is the operation of adding a sequence of numbers, and sigma notation is the shorthand that makes long sums writable: Σ from i = 1 to n of aᵢ is a₁ + a₂ + … + aₙ in a single expression. The subject’s power lies in the closed forms — formulas that evaluate whole families of sums without adding term by term — and in the convergence theory that says when an infinite sum has a finite value at all. This guide explains the notation, derives the arithmetic and geometric formulas, covers the standard power sums and the harmonic series, and states the convergence tests that distinguish a sum that settles from one that grows without bound.

Sigma notation: the grammar of sums

Sigma notation has four parts: the sigma itself, the index variable written below it with its starting value, the upper limit written above it, and the general term written after it. Σ from i = 3 to 6 of i² expands to 3² + 4² + 5² + 6² = 9 + 16 + 25 + 36 = 86. The index variable is bound inside the expression: changing i to j throughout changes nothing, just as renaming a variable inside a function changes nothing. The limits need not start at 1: Σ from k = 0 to 4 of 3ᵏ = 1 + 3 + 9 + 27 + 81 = 121, and a sum can run over any finite set of integers, including negative ones, provided the general term is defined at each.

The manipulation rules are the algebra of sums. A constant factor pulls out: Σ c·aᵢ = c·Σ aᵢ. A sum of sums splits: Σ (aᵢ + bᵢ) = Σ aᵢ + Σ bᵢ. A constant summed n times is n times the constant: Σ from i = 1 to n of 5 = 5n. Shifting the index — replacing i by i + 1 and adjusting the limits — leaves the sum unchanged, and is the standard trick for aligning two sums that must be combined. These rules are what make sigma notation a calculus rather than a mere abbreviation: sums can be manipulated symbolically before any number is substituted, which is how the closed forms below are derived.

Arithmetic series: pairing, and the quadratic formula

An arithmetic sequence adds a constant difference d at each step: a, a + d, a + 2d, …, a + (n − 1)d. Its sum is n/2 × (first + last) — the pairing argument: write the sum forwards and backwards, add the two lines term by term, and each pair totals first + last, giving n such pairs, hence twice the sum equals n × (first + last). The sum 1 + 2 + … + n is the special case with a = 1, d = 1: n(n + 1)/2. The sum of the first 100 integers is 100 × 101 / 2 = 5,050; the sum of the first 1,000 is 500,500. The formula is quadratic in n, which is why the sum grows roughly as n²/2 — doubling the count roughly quadruples the sum.

The pairing argument generalises to any arithmetic series, including decreasing ones (d negative) and ones with fractional differences. The sum of the even numbers 2 + 4 + … + 100 is 50/2 × (2 + 100) = 2,550. The sum of an arithmetic series is also the average term times the number of terms, where the average term is the mean of the first and last — a restatement of the same formula that is often the easier way to remember it. The calculator on this page applies the closed form where it recognises an arithmetic pattern.

Geometric series: the ratio, and the infinite sum

A geometric sequence multiplies by a constant ratio r at each step: a, ar, ar², …, arⁿ⁻¹. Its sum is a × (1 − rⁿ)/(1 − r) for r ≠ 1, derived by multiplying the sum by r, subtracting the two expressions, and observing that all but two terms cancel — the telescoping trick that is the geometric series’ signature. The sum 1 + 2 + 4 + 8 + 16 is (1 − 2⁵)/(1 − 2) = 31. The sum 1 + 1/3 + 1/9 + 1/27 is (1 − (1/3)⁴)/(1 − 1/3) = (80/81)/(2/3) = 40/27 ≈ 1.481.

When |r| < 1, the term rⁿ tends to zero as n grows, and the infinite series converges to a/(1 − r). The sum 1 + 1/2 + 1/4 + 1/8 + … is 1/(1 − 1/2) = 2: the partial sums 1, 1.5, 1.75, 1.875, … approach 2 without ever reaching it. When |r| ≥ 1, the infinite series diverges (except the trivial r = 1 case, where the sum is n·a and grows linearly) — the terms do not shrink, so the sum cannot settle. The geometric series is the model for compound interest, for present value, and for every process where each step multiplies the last: the same formula prices an annuity, sums a repeating decimal, and computes the total distance of a bouncing ball.

Power sums and the harmonic series

The sums of powers have closed forms of rising degree. The sum of the first n integers is n(n + 1)/2, quadratic. The sum of the first n squares is n(n + 1)(2n + 1)/6, cubic: 1 + 4 + 9 + 16 = 30, and 4 × 5 × 9 / 6 = 30. The sum of the first n cubes is [n(n + 1)/2]² — the square of the sum of the integers, a coincidence that surprises on first meeting: 1 + 8 + 27 + 64 = 100 = 10². These formulas are proved by induction, and they are the workhorses of discrete mathematics, appearing wherever a sum over a range must be evaluated exactly without iteration.

The harmonic series, 1 + 1/2 + 1/3 + 1/4 + …, diverges — it grows without bound, though very slowly: the partial sum after a million terms is only about 14.4, and reaching 100 would require about 10⁴³ terms. The divergence is proved by grouping: 1/3 + 1/4 > 2 × 1/4 = 1/2, 1/5 + … + 1/8 > 4 × 1/8 = 1/2, and each successive block of doubling size contributes more than 1/2, so the sum exceeds any bound. The harmonic series is the canonical example that terms tending to zero do not guarantee convergence: 1/n tends to zero, but not fast enough. The series 1/n², by contrast, converges — to π²/6, the Basel problem’s answer — because its terms shrink fast enough. The distinction between tending to zero and tending to zero fast enough is the whole content of convergence theory in one sentence.

Telescoping sums, and where summation is used

A telescoping sum is one where consecutive terms cancel: Σ (aᵢ − aᵢ₊₁) from i = 1 to n equals a₁ − aₙ₊₁, because every intermediate term appears once positive and once negative. The sum Σ 1/(i(i+1)) telescopes via the partial-fraction identity 1/(i(i+1)) = 1/i − 1/(i+1), giving 1 − 1/(n+1) = n/(n+1). Telescoping is the discrete analogue of the fundamental theorem of calculus — the sum of differences is the difference of the endpoints — and recognising a telescoping form is often the fastest route to a closed form, faster than any general formula.

Summation is used wherever a total is built from parts: the expected value of a random variable is a sum of values weighted by probabilities; the area under a curve is the limit of a sum of rectangles; the total interest on a loan is the sum of the interest portions of the payments; a signal is the sum of its frequency components. The calculator on this page evaluates finite sums from an expression and limits, and the closed forms in this guide provide the check: for any arithmetic or geometric sum, the closed form and the term-by-term addition must agree, and a disagreement means an entry error, not a mathematical mystery.

Frequently asked questions

What is sigma notation?

Σ (sigma) denotes a sum: Σ from i = 1 to n of aᵢ means a₁ + a₂ + … + aₙ. The index variable takes each integer value from the lower limit to the upper limit, the expression is evaluated at each, and the results are added.

What is the sum of the first 100 numbers?

5,050, by the arithmetic-series formula n(n+1)/2 = 100 × 101 / 2. The pairing argument: (1+100) + (2+99) + … gives 50 pairs each totalling 101.

What is the sum of a geometric series?

For first term a, ratio r, and n terms: a × (1 − rⁿ)/(1 − r). If |r| < 1 and the series runs forever, the sum converges to a/(1 − r). Example: 1 + 1/2 + 1/4 + … = 2.

Does the harmonic series converge?

No. 1 + 1/2 + 1/3 + … grows without bound, though very slowly — a million terms reach only about 14.4. Its terms tend to zero, but not fast enough for convergence, which is why the terms-tending-to-zero test is necessary but not sufficient.

What is a telescoping sum?

A sum where consecutive terms cancel, leaving only the first and last: Σ (aᵢ − aᵢ₊₁) = a₁ − aₙ₊₁. Example: Σ 1/(i(i+1)) from 1 to n equals n/(n+1). Recognising a telescoping form gives the closed form immediately.

How do you sum squares from 1 to n?

Use n(n+1)(2n+1)/6. For n = 10: 10 × 11 × 21 / 6 = 385. The sum of cubes is the square of the sum of the integers: [n(n+1)/2]².