<s>[INST] Evaluating the limit of $\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$ when $n\to\infty$ The following nested radical $$\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to $2$. 
We can consider a similar nested radical where the degree of the radicals increases: 
$$\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$$ 
which converges to a constant $C=1.8695973...$. Is there a closed-form expression for this limit? 
 [/INST] One of the well known formula by Indian mathematician Ramanujan is this one:
$$f(x)=x+1=\sqrt { \left( x+1 \right) ^{2}}=\sqrt {1+{x}^{2}+2\,x}=\sqrt {1+x\sqrt {(x+2)^2}}$$
$$=\sqrt {1+x\sqrt {1+{x}^{2}+4\,x+3}}=\sqrt {1+x\sqrt {1+ \left( x+1 \right) \sqrt {(x+3)^2}}}$$
$$
=\sqrt {1+x\sqrt {1+ \left( x+1 \right) \sqrt {1+ \left( x+2 \right)
\sqrt {1+ \left( x+3 \right) \sqrt {1+ \left( x+4 \right) \sqrt {\cdots}}}}}}
$$
So it is interesting to show that why your mentioned limit tends to  $f(0.86959730667536)$?
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