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I recently saw the identity $$\sqrt 2 = 2\left(1-\dfrac 13\right)\left(1+\dfrac 15\right)...$$

After some 'research' apparently this is derived from the cosine product expansion in this link.

My question is, how does one derive this formula? Thanks!

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1 Answer

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It follows from the Weierstrass Factorization Theorem (found here )

Basically, look at the Taylor expansion of $\cos (z)$. This is a polynomial, from the fundamental theorem of algebra, it can be represented as a product of its roots (sort of).

I'll be happy to elaborate more if you need, but this is enough to get you started on the right path.

Good luck and happy hunting!

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