Summation Calculator

What Is Sigma Notation?

Sigma notation — written with the Greek capital letter Σ — is a compact way to express the sum of a sequence of terms. Instead of writing 1 + 2 + 3 + 4 + 5, you can write Σ i for i = 1 to 5. The symbol Σ tells you to add up every value of the expression as the index variable steps from the lower bound to the upper bound.

Three parts make up any sigma expression: the lower bound (where the index starts, usually 1), the upper bound (where it stops), and the summand (the expression evaluated at each step). Understanding these parts lets you decode or construct any series sum in seconds.

How to Calculate the Sum of an Arithmetic Series

An arithmetic series is the sum of an arithmetic sequence — a list where each term increases or decreases by the same fixed amount, called the common difference d. The series 2 + 5 + 8 + 11 + 14 is arithmetic with a = 2 and d = 3.

The closed-form formula S_n = n/2 × (2a + (n−1)d) lets you compute the sum without adding every term individually. For the classic school problem of 1 + 2 + 3 + … + 100, plug in a = 1, d = 1, n = 100: S₁₀₀ = 100/2 × (2 + 99) = 50 × 101 = 5050. The young Carl Friedrich Gauss reportedly discovered this shortcut in seconds.

How to Calculate the Sum of a Geometric Series

A geometric series sums a sequence where each term is obtained by multiplying the previous term by a fixed ratio r. The series 2 + 6 + 18 + 54 + 162 is geometric with a = 2 and r = 3.

The formula S_n = a(1 − rⁿ)/(1 − r) applies whenever r ≠ 1. When |r| < 1 (a fraction), each successive term shrinks and the series converges — a series like 1 + 0.5 + 0.25 + 0.125 approaches a finite limit. When |r| > 1, the terms grow exponentially and the finite sum can become very large. When r = 1, every term equals a and the sum is simply n × a.

A special case: when |r| < 1 and n approaches infinity, the infinite geometric series converges to a/(1 − r). The series 1 + 1/2 + 1/4 + 1/8 + … converges to 1/(1 − 0.5) = 2.

Sum of Powers Formulas

Three identities cover the most common power series students encounter. These closed-form results are exact integers — no rounding or approximation involved.

Notice that the sum of cubes equals the square of the sum of integers — a beautiful identity that can be proved by induction: (1 + 2 + … + n)² = 1³ + 2³ + … + n³.

Worked Examples

Walking through examples from start to finish shows exactly how to apply each formula. Each calculation below follows the same three-step pattern: identify the parameters, apply the closed-form formula, and verify with a few individual terms.

Summation Calculator FAQ

What does Σ (sigma) mean in mathematics?

Σ is the Greek capital letter sigma, used in mathematics as a shorthand for summation. When you see Σ f(i) from i = a to b, it means "evaluate f(i) at every integer from a to b and add the results together." It is the discrete analogue of the integral sign ∫ used in calculus.

What is the formula for the sum of an arithmetic series?

The standard closed-form is S_n = n/2 × (2a + (n−1)d), where a is the first term, d is the common difference, and n is the number of terms. An equivalent form is S_n = n × (first term + last term) / 2, which is more intuitive: you average the first and last term, then multiply by the count.

How do I calculate the sum of a geometric series?

Use S_n = a × (1 − rⁿ) / (1 − r) when r ≠ 1, where a is the first term, r is the common ratio, and n is the number of terms. If r = 1, all terms are equal to a and the sum is simply n × a. If you need an infinite sum and |r| < 1, the series converges to a / (1 − r).

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

Σ i from 1 to n equals n(n+1)/2. For n = 100, that is 100 × 101 / 2 = 5050. This formula is attributed to Gauss, who discovered it as a child when his teacher asked the class to sum the integers from 1 to 100. He noticed that pairing 1 with 100, 2 with 99, and so on gives 50 pairs each summing to 101.

What is the difference between an arithmetic and a geometric series?

An arithmetic series has a constant difference between consecutive terms (e.g. 2, 5, 8, 11 — always add 3). A geometric series has a constant ratio between consecutive terms (e.g. 2, 6, 18, 54 — always multiply by 3). Arithmetic series grow linearly; geometric series grow (or shrink) exponentially depending on whether |r| is greater or less than 1.

Why does the geometric series formula break down when r = 1?

The formula S_n = a(1 − rⁿ)/(1 − r) divides by (1 − r). When r = 1, that denominator is zero, which is undefined. But the answer in this case is trivial: every term equals a, so the sum of n terms is simply n × a. This is why this calculator handles r = 1 as a special case with its own formula.