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In this answer, user Bill Dubuque mentioned a sufficient condition for the convergence of the infinite nested radical $\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots+\sqrt{a_n}}}}$

My question is whether there's a sufficient (and/or necessary) condition for the convergence of the infinite nested radical $\sqrt{a_1-\sqrt{a_2-\sqrt{a_3-\cdots-\sqrt{a_n}}}}$.

Any help will be appreciated. Thanks!

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1 Answers1

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Considering $a_{n}=1$, $\forall n$

$\sqrt{1}=1$,

$\sqrt{1-\sqrt{1}}=0$

$\sqrt{1-\sqrt{1-\sqrt{1}}}=1$

$\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1}}}}=0$

This oscillates constantly, so it doesn't converge.

(Unless you force the last term $a_{n}\in (0,1)$ the limit will tends to $\frac{\sqrt{5}-1}{2}$.)

Probably $0<a_{n}<a_{n-1}^2$ is sufficient enough for convergence.

Ng Chung Tak
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