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Ramanujan's infinite nested radical states that $$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}=3.$$

Now instead, consider the following infinite nested radical $$\sqrt{1+p_1\sqrt{1+p_2\sqrt{1+p_3\sqrt{1+\cdots}}}}$$ where $p_n$ represents the $n$-th prime. How to solve this infinite nested radical?

user51819
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    There is no reason to believe there is a closed form for this. – GEdgar May 05 '16 at 13:35
  • The limit is near 3.22568498344851051856982775582317685533646416143213032426502169860927386459508390331119659768856929195159856682320561985158638328820199852394889053547987785793846478974996979088874003139014155342428713640693462247154958736855855470365471911538525967656663540343097162017154067171624676290223851165722937350851664632701780796912864265789241498492714034021555834084999829017955370529051821851669929889207597 if this helps... – Raymond Manzoni May 05 '16 at 13:44
  • Out of curiosity how did you get that Raymond? – Ian Miller May 05 '16 at 13:49
  • @Ian Miller: Well a short pari/gp script : s=0;for(k=-2000,-1,s=sqrt(1+prime(-k)s));s (to 400 digits with \p 400) or more nicely : f(m)=local(s=0);forstep(k=m,1,-1,s=sqrt(1+prime(k)s));s – Raymond Manzoni May 05 '16 at 13:55

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