calculus and analysis - Limit of Nested Radical $sqrt{1+sqrt[2!]{2^2+sqrt[3!]{3^3+...}}}$

I want to find limit of infinite nested radical

$\quad \quad \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$

but I don't know how to define this expression in Mathematica. How can I define it and find the limit:

$\quad \quad \lim_{n \to\infty} \sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...+\sqrt[n!]{n^n}}}}$

Answer

Here is an approach using FixedPoint, where I keep the output in exact form to see how many terms are needed to satisfy a given tolerance:

Clear[x];

step[{n_, f_}] := {n + 1, f /. x -> (n^n + x)^(1/n!)};
tolerance = $MachineEpsilon;
sum =
 Last@FixedPoint[step, {2, Sqrt[1 + x]}, 
    SameTest -> (tolerance > Abs[Last[#1] - Last[#2]] /. 
        x -> 0 &)] /. x -> 0

$$\sqrt{1+\sqrt{4+\sqrt[6]{27+\sqrt[24]{256+\sqrt[120]{31 25+\sqrt[720]{46656+\sqrt[720]{7}}}}}}}$$

N[sum, 16]

$1.843075984668544$

tolerance = 10^-40;
sum =
 Last@FixedPoint[step, {2, Sqrt[1 + x]}, 
    SameTest -> (tolerance > Abs[Last[#1] - Last[#2]] /. 
        x -> 0 &)] /. x -> 0

In FixedPoint, the function step handles both the index n and the iterated root f. The dummy variable x is used to insert the next root.

Comments

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