User:Hirak 99/MathProblems

Ramanujan's Problem
Find $$\sqrt{1+2 \sqrt{1 + 3 \sqrt{1 + 4\sqrt{1+...}}}}$$.

Proposed solution
Let $$f_n(x)=\sqrt{1+x \sqrt{1 + (x+1) \sqrt{...(x+n-1)\sqrt{1+(x+n)(x+n+2)}}}}$$

Let $$f(x)=\lim_{n \to \infty}f_n(x)=\sqrt{1+x \sqrt{1 + (x+1) \sqrt{1+(x+2)\sqrt{...}}}}$$, if it exists.

Now of course, the limit exists since $$f_n(x)=x+1 \forall n$$, and it trivially equals $$f(x)=x+1 \,$$

So $$\sqrt{1+2 \sqrt{1 + 3 \sqrt{1 + 4\sqrt{1+...}}}}=3$$