A question about a limit involving nested radicals

Question

$$\lim\limits_{n\rightarrow \infty}1＋\sqrt{2+\sqrt[3]{3＋…\sqrt[n]{n}}}$$

Any hint will be appreciated

Answer 1

The limit exist (since the sequence is increasing and bounded), but it has probably no close form.

By matlab,

function [f]=fxy(x,y)

if x==1
    f=y^(1/y);
else
    f=(y+fxy(x-1,y+1))^(1/y);
end

and

format long

for n=1:1:20
    f(n)=fxy(n,1);
end

plot(f,'o')

give the graph :

and we have $$\lim_{n\to\infty}f(n,1)\approx2.911639216245824$$

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