How does one prove the following limit?
limn→∞√1+2√1+3√1+⋯√1+(n−1)√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+x√a(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 ¯limn→∞logan2n <∞.
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