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2 votes

Cosine as nested roots

Let $r$ be a nonzero integer. $\cos(2\pi/r)$ is the real part of $e^{2\pi i/r}$, which is a root of the polynomial $x^r-1$. It is known that the Galois group of this polynomial is abelian, which ...
Gerry Myerson's user avatar
2 votes

Evaluating $ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $

Another comment, not an answer. I have listed below the ‘primitive’ forms with all positive terms and the first term as small as possible, satisfying $${I_n}^2=an^2+bn+c+(d+en)\sqrt{I_{n+1}}$$ $$2=\...
Paul vdVeen's user avatar
1 vote

Evaluating $ \sqrt{1+4\sqrt{1+9\sqrt{1+16\sqrt{1+\cdots}}}} $

This is a comment not an answer. This will be tricky to find a closed form for. To see why let us consider a slight generalization of the ideas in @Chickenmancer's link. For any $x,n,p$ we have that: $...
Sidharth Ghoshal's user avatar

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