Ramanujan's nested radical is great!

Algebra Level 4

Let \( S= \displaystyle\sum_{x=0}^{62}\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}\)

Hence, find \(\lfloor S\rfloor\)

Hints: \(3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}\)

\(2+1=\sqrt{1+2\sqrt{1+(2+1)\sqrt{1+(2+2)\sqrt{1+\cdots}}}}\)

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