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How to compute $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}$$ & $$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}$$?

I understand that $$\sqrt{(1\sqrt{(2\sqrt{(3\dots)})})}=(1^{1/2})(2^{1/4})(3^{1/8})\cdots$$

and

$$\sqrt{(1+\sqrt{(2+\sqrt{(3+\cdots)})})}=(1+(2+(3+(\cdots))^{1/8})^{1/4})^{1/2} $$

How to Proceed further?

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    If *compute* has the sense of *determine the value of*, I see no reason why this should be possible (do you have any?). On the other hand, one can show that these iterative definitions indeed converge (is this in fact your question?).2012-09-23
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    I understand that the above expressions converge.But I want to know how to determine the value of above expressions.2012-09-23
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    Then we are back to my first question: do you have any reason to suspect that one can determine their value?2012-09-23
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    Yes. Because both the above expressions converge(to my knowledge). If we take the first expression though the magnitude of the terms increase with deeper nesting so does the power which keeps on decreasing i.e., 4th root, 8th root, etcetera2012-09-23
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    The same applies to $\sum\limits_{n\geqslant1}10^{-n!}$, which has no known expression.2012-09-23
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    @did: Your comments would be more constructive if you yourself had some reason to believe that this was similar in form to other expressions that could or could not not be evaluated in closed form. Your comments would be more well defined if you talked about "closed form in terms of ..." (with ... being some specified constants and operations) rather than "known expression."2012-09-24
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    @BenCrowell Surely you noticed that the example I provided fulfills the conditions Ranjan presented as arguments for the existence of closed forms. // Unfortunately, I fail to see the net mathematical input your comment represents, hence you might want to explain what your point is exactly.2012-09-24

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