Tool to perform formal calculations of products with the operator ∏ (uppercase pi), allowing infinite products or defined arithmetic multiplication from 1 to n.
Product ∏ - dCode
Tag(s) : Arithmetics
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!
In mathematics, the product, denoted $ \prod $, is the result of the multiplication $ \times $ of a series of numbers.
In arithmetic, the product notation $ \prod_1^n $ allows you to calculate a finite multiplication repeated from $ 1 $ up to $ n $ with an increment of 1 (unless otherwise stated).
Example: The product of the first $ 4 $ integers (also called factorial of 4) is written $$ 1 \times 2 \times 3 \times 4 = \prod_{i=1}^{4} i = 4! = 24 $$
Example: The increasing factorial (or Pochhammer function) is written $ (x)_n=x(x+1)(x+2)\cdots(x+n-1) $ and $ (x+1)_n=(x+1)(x+2)\cdots(x+n) = \prod_{k=1}^{n} (x+k) $
Sometimes the product does not converge towards a value, it can diverge and have no result.
The notation $ \prod_1^\infty $ (sometimes shortened to $ \prod $) indicates the calculation of an infinite multiplication repeated from $ 1 $ to infinity $ \infty $ by an increment of 1 (unless otherwise stated).
Example: $$ \prod_{n=2}^{\infty} \left( 1-\frac{1}{n^2} \right) = \frac{1}{2} $$
Some functions can be described as an infinite product
Example: The sine function $$ \sin{x} = x \prod_{n=1}^\infty \cos\left( z/2^n \right) $$
Example: The logarithm function $$ \ln{x} = (x-1) \prod_{n=1}^\infty \frac{2}{1+x^{1/2^n}} $$
The notation $ \prod \prod $ is read $ \prod \left (\prod \right) $ so the inner product (between the parenthesis) is calculated first, then the outer product is calculated.
dCode retains ownership of the "Product ∏" source code. Any algorithm for the "Product ∏" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Product ∏" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Product ∏" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
The content of the page "Product ∏" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Product ∏ on dCode.fr [online website], retrieved on 2025-04-24,