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Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio and $U(x) \sim x$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to convergence. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see the preprint Constructive proof of Herschfeld's Convergence Theorem, arXiv:1907.02700 by Ran Gutin

Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio and $U(x) \sim x$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to converge. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see the preprint Constructive proof of Herschfeld's Convergence Theorem, arXiv:1907.02700 by R Gutin

Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio and $U(x) \sim x$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to convergence. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see this preprint: https://arxiv.org/pdf/1907.02700.pdf

Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio and $U(x) \sim x$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to convergence. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see the preprint Constructive proof of Herschfeld's Convergence Theorem, arXiv:1907.02700 by Ran Gutin

Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to convergence. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Finally, and perhaps not so importantly, there is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical.

Consider the function which is the limit of the sequence $x^{2^0}, \sqrt{1 + x^{2^1}}, \sqrt{1 + \sqrt{1 + x^{2^2}}}, \dotsc$ Call this function $U(x)$, with domain $\{x \in \mathbb R \mid x \geq 0\}$. Notice that for $x \in [0,1]$, $U(x)$ is equal to the golden ratio. It can be shown that for $x > 1$, $U(x)$ is greater than the golden ratio and $U(x) \sim x$. Finally, this function can be used to show that problem (I) is equivalent to problem (II).

Problem (I): Find the limit, and upper bound on convergence rate, for the sequence $\sqrt{u_1}, \sqrt{u_1 + \sqrt{u_2}}, \sqrt{u_1 + \sqrt{u_2 + \sqrt{u_3}}}$, etc. where all $u_n$ are non-negative.

Problem (II): Find a way to compute the terms of the sequence $n \mapsto \sup_{k \geq 0} u_{n+k}^{2^{-(n+k)}}$.

This is a "constructive" analogue of Herschfeld's Convergence Theorem. Herschfeld's theorem gives a necessary and sufficient condition for an infinite radical to convergence. The above gives a necessary and sufficient condition for being able to compute what an infinite radical converges to. The two necessary and sufficient conditions are quite similar.

Postscript: There is a sense in which $U(x)$ is a family of transfinite radicals, which generalises the notion of an infinite radical. For more, see this preprint: https://arxiv.org/pdf/1907.02700.pdf