Sunday, 28 April 2013

real analysis - Evaluating the nested radical sqrt1+2sqrt1+3sqrt1+cdots.



How does one prove the following limit?
limn1+21+31+1+(n1)1+n=3.



Answer



This is the special case  x,n,a=2,1,0  in Ramanujan's second notebook, chapter XII, entry 4:



x+n+a = ax+(n+a)2+xa(x+n)+(n+a)2+(x+n)



Below is Ramanujan's solution of the given special case - which was submitted to a journal in April 1911. Note that his solution is incomplete (exercise: why?). For further discussion see this 1935 Monthly article, Herschfeld: On infinite radicals. It also appeared as Problem A6 on the 27th Putnam competition, 1966. Vijayaraghavan proved that a sufficient criterion for the convergence of the following sequence  a1+a2++an   is that   ¯limnlogan2n <. 



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