Sunday, 28 April 2013

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



How does one prove the following limit?
lim



Answer



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



\rm x + n + a\ =\ \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n) \sqrt{\cdots}}}



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 \ \sqrt{a_1 + \sqrt{a_2 +\:\cdots\: +\sqrt{a_n}}}\ \ is that \rm\displaystyle\ \ {\overline \lim}_{n\to\infty}\frac{\log{a_n}}{2^n}\ < \infty\:.\



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