Infinite products of 2's and divisions by 2's converging to 2\sqrt{2}?

I was just doing a problem when a weird idea occurred to me. The problem required me to think about an infinite multiplications and divisions by 2:

222222...222222...\dfrac{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 ...}{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 ...}

Which I instinctively assumed was 1. However, I quickly realized this was probably wrong, since it led to some contradictions within the problem. The "crazy" idea that occurred to me is that perhaps this converges to 2\sqrt{2} under some sort of Cesaro convergence for products, which makes total sense within the context of the problem. This is because we can express it as follows:

n=021n=212121212121...=211+11+11+11+...\huge\prod_{n=0}^{\infty}2^{-1^{n}}=2^1\cdot2^{-1}\cdot2^1\cdot 2^{-1}\cdot2^1\cdot 2^{-1}...=2^{1-1+1-1+1-1+1-1+...}

Where we can recognize Grandi's sum which is cesaro convergent to 12\dfrac{1}{2}, and thus:

n=021n ’Cesaro’ converges to 212=2 ?\huge\prod_{n=0}^{\infty}2^{-1^{n}}\text{ 'Cesaro' converges to } 2^{\frac{1}{2}}=\sqrt{2}\text{ ?}

Please enlighten me.

Note by Saúl Huerta
1 month, 3 weeks ago

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1 vote

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This is meaningless, like n=1n=112\displaystyle \sum_{n=1}^\infty n =-\dfrac{1}{12}. It has to do with Riemann's ζ\zeta function.

Edward Christian - 1 month, 2 weeks ago

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The Grandi Series is a way of assigning a divergent series a finite sum, thus the above explanation or rather apparent proof is subject to the flaws that if infinty is an even entity then the product converges to 1 else if it isn't then the series converges to 2.

Sarthak Sahoo - 1 month, 3 weeks ago

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But then how can we assign it a finite product? As I stated, for my purposes I find that 2\sqrt{2} works well.

Saúl Huerta - 1 month, 2 weeks ago

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you can assign it a finite product in the sense of cesaro convergence, however this only works when you want to analytically expand the domain of a function. However if the question was pertaining to real analysis then the above infinite product is undefined.

Sarthak Sahoo - 1 month, 2 weeks ago

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