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Summation Σ

Tool to perform formal calculations with the summation operator Σ ∑ (sigma), allowing arithmetic additions from 1 to n.

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Summation Σ -

Tag(s) : Arithmetics

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# Summation Σ

## Double Sum Σ Σ Calculator

### What is a sum ∑? (Definition)

In mathematics, the summation, denoted $\sum$, is the result of the addition of a series of numbers.

The symbol ∑ is called the sum operator, it is an addition calculator (finite or infinite).

### How to calculate a finite sum?

In arithmetic, the summation notation $\sum_1^n$ (with the Greek letter sigma uppercase) allows to compute a finite addition going from $1$ to $n$ with an increment of 1 (by default).

Example: The sum of the first $5$ integers $$1 + 2 + 3 + 4 + 5 = \sum_{i=1}^{5} i$$

Sometimes the sum can be simplified with a formula:

Example: The sum of the $n$ first integers $$1 + 2 + 3 + \cdots + (n-1) + n = \sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$

The calculation by hand being time-consuming, some sums are interesting to learn/know.

Sometimes the sum does not converge to a value, it can diverge and not have a formula to calculate it.

### How to calculate an infinie de sum?

The notation $\sum_1^\infty$ (sometimes shortened in $\sum$) indicates the computation of an infinite addition going from $1$ to infinity $\infty$ with an increment of 1 (by default).

Example: The sum of the inverses of the $n$ prime squares (Basel problem) $$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

The demonstration of these sums often involves a limit calculation or a series expansion.

### How to calculate a double summation?

The notation $\sum \sum$ is read $\sum \left( \sum \right)$ so the inner sum (inside the parenthesis) is calculated first first, before the outer sum is calculated in a second step.

### How to make the Σ sum symbol?

The summation is written with the dedicated mathematical operator ∑ (Unicode U+2211) inspired from the Greek letter sigma uppercase Σ (Unicode U+03A3).

In Greek, sigma corresponds to the letter S (like the first letter of Sum).

In LaTeX, the operator is \sum

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