The definition of an **infinite product** is very similar to the definition of an infinite sum. Instead of adding an infinite number of terms, you’re multiplying. For example, 1·2·3·,…,∞.

More formally, let {x_{n}} represent a numerical series. The infinite product of the numbers x

And the n*th* partial product is:

At first glance, the two formulas may seem identical. However, note the terminating x_{n} in the second formula, indicating the end of the partial sum.

We can define **convergence **for infinite products in a couple of different ways. The product is convergence if either

Infinite sums and products are widely used in applied mathematics and analysis as well as number theory.

## Infinite Product Examples

One of the earliest infinite products is the **Wallis formula**, gives an expansion for π as an infinite product:

Which can be written, equivalently, in product notation:

The **Euler formula** is another famous infinite product. It is defined as [3]:

Where:

- p
_{n}is the n*th*prime, - The right hand side is the Riemann zeta function.

## Infinite Product Formula Example

Expressing a function as an infinite product makes it easier to see where the zeros of a function are. As a simple example, the polynomial function f(x) = x – x^{3} can be factored as a product:

f(x) = x – x^{3} = x(1 – x^{2}) = x(1 – x)(1 + x)

This makes it clearer that the zeros are at x = 0, x = 1, or x = -1.

In the same way, sin(πx) can be rewritten as an infinite product formula [4]:

## References

[1] Mahmudov, E. (2013). Single Variable Differential and Integral Calculus. Atlantis Press.

[2] Kuratowski,, K. & Lohwater, A. (2014). Introduction to Calculus, Elsevier Science p.74.

[3] 2.6. Infinite Products. Retrieved August 17, 2021 from: https://www.math.uh.edu/~shanyuji/Complex/complex-1/cx-11-online.pdf

[4] Smith, M. Math 1320 Lab EC. Retrieved August 17, 2021 from: http://www.math.utah.edu/~msmith/pastteaching/1320/labEC.pdf