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About $\sqrt[k]{l + \sqrt[k]{l + \sqrt[k]{l + ...}}} $ asymptotics
Consider simple nested radicals
More precisely Let
$$ K > 1 , 1 \leq l $$
$$X(j,K) = X_\infty(j,K)$$
$$X_0(j,K) = a(j,K)$$
$$X_n(j,K) = \sqrt[k]{j + X_{n-1}(j,K)}$$
$$Y(j,K) = \frac{j + X_{\...
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Closed form for $\sqrt{1+\sqrt{x+\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+...}}}}}$
Inspired by this question that I recently saw, I was wondering if there is a closed form for $$y = \sqrt{1+\sqrt{x+\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+...}}}}}$$
as a function of $x$.
Usually, in problems ...
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What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?
Let's look at the continued radical
$ R = \sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $
whose signs are defined as $ (+, -, +, -, -, + ,-, -, -,...)$, similar to the sequence $...
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