evaluating $ \sqrt {2+ \sqrt {3+ \sqrt{4+ \sqrt{5+\cdots}}} }$

Question

How do we evaluate the infinite nested radical $ \sqrt {2+ \sqrt {3+ \sqrt{4+ \sqrt{5+\cdots}}} } $ $\space $?

Please help

N.B. :- It is not a duplicate

Answer 1

There is no known closed form for this number, except $C^2-1$, where $C=\sqrt{1+\sqrt{2+\sqrt{3+\dots}}}$ is the Nested Radical Constant. By a numerical evaluation, I find

$$C^2-1=2.090327576790576359192544506688116904296\dots$$