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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

## Sum ∑ Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

## Double Sum Σ Σ Calculator

 Result Format Automatic Selection Exact Value (when possible) Approximate Numerical Value Scientific Notation

### 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), it allows you to shorten the writing of multiple + (plus).

### 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 is 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.

### What is the list of mathematical series to know?

There are many mathematical series (finite or infinite) useful to learn and know, here is a non-exhaustive list:

— Faulhaber's formulas (sum of p-th powers of the first m integers):

$$\sum_{k=1}^m k = 1 + 2 + \cdots + m = \frac{m(m+1)}{2}$$

$$\sum_{k=1}^m k^2 = \frac{m(m+1)(2m+1)}{6} = \frac{m^3}{3}+\frac{m^2}{2}+\frac{m}{6}$$

— The particular values of the Riemann Zeta function:

$$\sum^{\infty}_{k=1} \frac{1}{k^2} = \zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$$

$$\sum^{\infty}_{k=1} \frac{1}{k^4} = \zeta(4) = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90}$$

— Powers and exponentials

$$\sum^{\infty}_{k=0} \frac{1}{k!} = \frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots = e$$

$$\sum_{k=0}^{n} z^k = \frac{1-z^{n+1}}{1-z}$$

$$\sum_{k=0}^\infty \frac{z^k}{k!} = e^z$$

$$\sum_{k=0}^\infty k \frac{z^k}{k!} = z e^z$$

— Trigonometric functions

$$\sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!} = \sin(z)$$

$$\sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!} = \cos(z)$$

$$\sum_{k=0}^n {n \choose k} = 2^n$$

$$\sum_{k=0}^\infty {\alpha \choose k} z^k = (1+z)^\alpha , \quad |z|<1$$

$$\sum^{\infty}_{k=1} \frac{(-1)^{k+1}}{k} = \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots = \ln(2)$$

$$\sum^{\infty}_{k=1} \frac{(-1)^{k+1}}{2k-1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots = \frac{\pi}{4}$$

### How to calculate a double summation (nested sums)?

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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Summation Σ on dCode.fr [online website], retrieved on 2024-09-13, https://www.dcode.fr/summation-calculator

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