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Infinite Product

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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 {xn} represent a numerical series. The infinite product of the numbers xn, n = 1, 2, 3, … is defined as [1]:
infinite product formal definition

And the nth partial product is:
nth partial product

At first glance, the two formulas may seem identical. However, note the terminating xn 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

  1. The limit of an infinite sum is finite and not zero, [1] or.
  2. The nth partial sum converges [2].

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:
wallis formula

Which can be written, equivalently, in product notation:
wallis formula product notation

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

Where:

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 – x3 can be factored as a product:
f(x) = x – x3 = x(1 – x2) = 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]:
sin product formula


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

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Stephanie Glen. "Infinite Product" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/infinite-product/
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