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I was solving a problem, asking me to prove the identity $3=\sqrt{1+2\sqrt{1+3\sqrt{1+4...}}}$ which was first posed by Ramanujan.

The standard answer goes as $$3=\sqrt{1+2\cdot4}=\sqrt{1+2\sqrt{1+3\cdot5}}=\dots=\sqrt{1+2\sqrt{1+3\sqrt{1+4...}}}$$ which I think can be stated rigorously as "Let $$x_n=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\dots\sqrt{1+(n\cdot n+2)}}}}}$$ then $x_n=3$ for every $n$."

However, I interpret this problem to be "Let $y_n=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\dots\sqrt{1+n}}}}}$, then $\lim_{n\to\infty}y_n=3$."

It is obvious that $y_n\le3$ and is increasing, therefore $\lim_{n\to\infty}y_n\le3$. I believe that they are actually equal, but have trouble estimating the difference $|3-y_n|$. Does anyone have any idea?


marked as duplicate by Hans Lundmark, rtybase, Sahiba Arora, Jean-Claude Arbaut, Adrian Keister Aug 18 '18 at 0:21

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