What is summation notation (Σ)?

Summation notation, written with the Greek letter Σ (sigma), is a compact way to express the addition of a sequence of numbers. For example, Σ from n=1 to 5 of n means 1+2+3+4+5 = 15. It is widely used in mathematics, statistics, and engineering.

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

The sum of an arithmetic series is S = n/2 × (2a₁ + (n−1)d), where a₁ is the first term, d is the common difference, and n is the number of terms. For example, the sum of 1+3+5+7+9 (a₁=1, d=2, n=5) is 5/2 × (2 + 8) = 25.

What is the formula for the sum of a geometric series?

The sum of a finite geometric series is S = a₁(1−rⁿ)/(1−r), where a₁ is the first term, r is the common ratio, and n is the number of terms (when r ≠ 1). For an infinite geometric series where |r| < 1, the sum converges to S = a₁/(1−r).

What is the sum of 1 to n?

The sum of integers from 1 to n is given by the formula S = n(n+1)/2. For example, the sum from 1 to 100 is 100 × 101 / 2 = 5,050. This is often called Gauss's formula.

What is the sum of n squared (Σ n²)?

The sum of the squares of the first n integers is S = n(n+1)(2n+1)/6. For example, Σ n² from n=1 to 5 is 1+4+9+16+25 = 55, which equals 5×6×11/6 = 55.

Free Summation Calculator (Σ) — Series & Sigma Notation

Q: What is the sum of 1 to n?

The sum of integers from 1 to n is given by the formula S = n(n+1)/2. For example, the sum from 1 to 100 is 100 × 101 / 2 = 5,050. This is often called Gauss's formula.

Q: What is the sum of n squared (Σ n²)?

The sum of the squares of the first n integers is S = n(n+1)(2n+1)/6. For example, Σ n² from n=1 to 5 is 1+4+9+16+25 = 55, which equals 5×6×11/6 = 55.

What is a Summation Calculator?

A summation calculator (or sigma calculator) computes the total of a sequence of numbers defined by a formula or pattern. In mathematics, summation is written using the Greek capital letter Σ (sigma), and is one of the most fundamental operations in calculus, statistics, and algebra. For example:

Read as: "sum of n, from n=1 to 5" Σ (n=1 to 5) n = 1 + 2 + 3 + 4 + 5 = 15

Our free summation calculator handles four types of series: simple integer ranges, arithmetic series (constant difference), geometric series (constant ratio), and fully custom formulas — so you can evaluate expressions like Σ n², Σ 1/n, or Σ (2n+1) without any manual working.

How to Use the Summation Calculator

Step 1 — Choose a Mode

Select the type of series you want to sum: Range, Arithmetic, Geometric, or Custom Formula.

Step 2 — Enter Your Values

For a range, enter the start and end integers. For arithmetic or geometric series, enter the first term and the common difference or ratio, plus the number of terms. For a custom formula, type your expression using n as the variable (e.g. n^2, 2*n+1, 1/n).

Step 3 — Click Calculate

The calculator instantly shows the total sum (Σ), number of terms, average term value, min/max, and a full list of the first 20 terms in the sequence so you can verify the result.

Summation Formulas Reference

Series Type	Formula	Example
Sum of integers 1 to n	n(n+1)/2	Σ 1→100 = 5,050
Sum of squares (Σ n²)	n(n+1)(2n+1)/6	Σ n² 1→10 = 385
Sum of cubes (Σ n³)	[n(n+1)/2]²	Σ n³ 1→5 = 225
Arithmetic series	n/2 × (2a₁ + (n−1)d)	1+3+5…(10 terms) = 100
Geometric series (finite)	a₁(1−rⁿ)/(1−r)	1+½+¼… (8 terms) ≈ 1.992
Geometric series (infinite, \|r\|<1)	a₁/(1−r)	1+½+¼+… = 2

Tip: Our custom formula mode lets you type any of these formulas directly. Enter n^2 for Σ n², n^3 for Σ n³, or n*(n+1)/2 for triangular numbers.

Common Summation Examples

Sum of 1 to 100 (Gauss's Problem)

The famous problem solved by Carl Friedrich Gauss as a schoolboy: the sum of all integers from 1 to 100. Using the formula n(n+1)/2 = 100×101/2 = 5,050. Try it with Mode: Range, Start: 1, End: 100.

Σ n² (Sum of Squares)

The sum of the squares of the first n integers. For n=10: 1+4+9+16+25+36+49+64+81+100 = 385. In our calculator: Custom Formula, enter n^2, Start: 1, End: 10.

Geometric Series (½ + ¼ + ⅛ …)

A geometric series with first term 1 and ratio 0.5. As you add more terms it approaches (but never reaches) 2 — this is convergence in action. Try: Geometric mode, a₁=1, r=0.5, and increase the number of terms to see the sum approach 2.

Sum of Odd Numbers

The sum of the first n odd numbers always equals n². The first 5 odd numbers: 1+3+5+7+9 = 25 = 5². Enter Custom Formula 2*n-1, Start: 1, End: 5 to verify.

Why Use Our Sigma Calculator?

📐 4 Modes

Range, arithmetic, geometric, and custom formula — covers every standard series type in one tool.

🔣 Custom Formulas

Type any expression in n: n^2, 1/n, sqrt(n), n*(n+1) and more.

📋 Term Breakdown

See the first 20 individual terms listed so you can verify the sequence, not just the final sum.

🆓 Free Forever

No sign-up, no subscription, no limits. Use it as many times as you need.

Frequently Asked Questions

What is Σ (sigma) in maths?

Σ is the Greek capital letter sigma, used in mathematics to represent summation — the process of adding up a sequence of numbers according to a rule or formula. It is read as "the sum of" and always comes with a starting index, ending index, and expression to evaluate.

What is the sum of 1 + 2 + 3 + ... + n?

The sum of all integers from 1 to n is given by Gauss's formula: S = n(n+1)/2. So the sum from 1 to 10 is 55, the sum from 1 to 100 is 5,050, and the sum from 1 to 1,000 is 500,500.

What is an arithmetic series?

An arithmetic series is a sequence where each term increases by a fixed amount called the common difference (d). For example 2, 5, 8, 11, 14 has a common difference of 3. The sum of n terms is S = n/2 × (2a₁ + (n−1)d), where a₁ is the first term.

What is a geometric series?

A geometric series is a sequence where each term is multiplied by a constant called the common ratio (r). For example 3, 6, 12, 24, 48 has a ratio of 2. If |r| is less than 1, the series converges to a finite sum even with infinitely many terms.

What formulas can I enter in custom mode?

You can use any expression involving n and standard maths: + - * / ^ (power), sqrt(), abs(), log(), ln(), sin(), cos(), floor(), ceil(), and the constants pi and e. For example: n^2 + 2*n + 1, 1/(n^2), sqrt(n).

Does a geometric series always converge?

No. A geometric series only converges (reaches a finite sum as you add infinite terms) when the absolute value of the common ratio is less than 1 (i.e. |r| < 1). If |r| ≥ 1, the series diverges — the sum grows without bound.

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