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Suppose I want to slow down the convergence speed of a given infinite product. As example, suppose we are working with the following well-known infinite products:

  • $\cos (x) = \displaystyle \prod_{n = 1}^{\infty} \Bigg[ 1 - \frac{4x^2}{\pi^2 (2n - 1)^2} \Bigg]$,

  • $\gamma (z) = \Bigg[z e^{z\gamma} \displaystyle \prod_{r = 1}^{\infty} \Bigg( 1 + \frac{z}{r} \Bigg) e^{-z/r} \Bigg]^{-1}$,

  • $\displaystyle \frac{z+m-1}{z+m} = \prod_{n = m}^{\infty} \frac{(n+z-1)(n+z+1)}{(n+z)^2}$, for $0 \neq z + m$

In the plot below, I show an example of convergence for first infinite product :

$\cos(0.4\pi)$ convergence

These products converge too fast. Is there a way we can modify these infinite products (or any other infinite product, or even a finite product - until a given iteration $n$, let us say), by adding a parameter for slowing down their convergence speeds? Making this question a more general one: Is there any infinite product out there, which can control both the speed of convergence and its final value?

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