Entry 19

The value of the infinitely nested radical $$F(m)=\sqrt[m]{1+\sqrt[m]{1+\sqrt[m]{1+\sqrt[m]{1+\dots}}}}$$ for integer $m>1$ is well-known to be an algebraic number of degree $m$. For $m=2$, it is the golden ratio while $m=3$ yields the plastic constant. Define $$G(n)=\sqrt[n]{1+2\sqrt[n]{1+3\sqrt[n]{1+4\sqrt[n]{1+\dots}}}}$$ For degree $n=2$, Ramanujan found that it was simply $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots}}}}$$ More generally $$x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}$$ To see this, note that $$x+1=\sqrt { \left( x+1 \right) ^{2}}=\sqrt {1+{x}^{2}+2\,x}=\sqrt {1+x\sqrt {(x+2)^2}}\\=\sqrt {1+x\sqrt {1+{x}^{2}+4\,x+3}}=\sqrt {1+x\sqrt {1+ \left( x+1 \right) \sqrt {(x+3)^2}}} =\dots$$ However, closed-forms for higher $n$ such as $G(3) \approx 1.702219132695458$ are not known.