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

Probably not much help, but related: http://math.stackexchange.com/questions/929156/nested-radicals-induction Still, you might have a look at other questions with the nested-radicals tag. — Gerry Myerson, Nov 07 '16 at 12:16
I can show that the sequence converges and give an upper estimate of the limit, but no idea about the exact value of the limit. — Julián Aguirre, Nov 07 '16 at 13:24
Duplicate: $\displaystyle\lim_{n\to\infty}1+\sqrt{\vphantom{3}2+\sqrt[3]{\vphantom{H}3+\cdots+\sqrt[n]{\vphantom{3}n}}}$. (Found using Approach0.xyz) — Workaholic, Nov 07 '16 at 14:49

score 0 · Answer 1 · answered Nov 07 '16 at 13:05

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

1 Answers1