Search for a tool
Summation Σ

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

Results

Summation Σ -

Tag(s) : Arithmetics

Share
Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Summation Σ' tool for free! Thank you!

Summation Σ

Sum ∑ Calculator










Double Sum Σ Σ Calculator












Answers to Questions (FAQ)

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

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

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

Binomial coefficients

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

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

Harmonic series

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

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

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

Source code

dCode retains ownership of the "Summation Σ" source code. Except explicit open source licence (indicated Creative Commons / free), the "Summation Σ" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Summation Σ" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Summation Σ" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Summation Σ" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Summation Σ on dCode.fr [online website], retrieved on 2024-12-03, https://www.dcode.fr/summation-calculator

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Summation Σ' tool for free! Thank you!


https://www.dcode.fr/summation-calculator
© 2024 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
 
Feedback