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The following beautiful formula was discovered by S.Ramanujan  [1] , [2] , [3, Question 289 and Solution, p. 323]

FormBox[Cell[TextData[{Cell[BoxData[Sqrt[1 + 2 Sqrt[1 + 3 Sqrt[1 + 4 Sqrt[1 + 5 Sqrt[1 + ...]]]]] = 3]], .}]], TraditionalForm]

To see the speed of the convergence go to

It appeared as a problem posed by him asking what is value of the left hand side infinite nested continued root. When after several months no answer was supplied, he presented a more general formula

x + n + a = √ (a x + (n + a)^2 + x (a(x + n) + (n + a)^2 + ... + (x + n) (a(x + 2 n) + (n + a)^2 + (x + 2 n) ...^(1/2))^(1/2))^(1/2))(1)

To see this note that if typeset structure denotes the right hand side of  (1) then typeset structure satisfies the functional equation typeset structure which is solved by typeset structure.  For typeset structure and typeset structure we get

x + 1 = (1 + x (1 + (1 + x) (1 + ...)^(1/2))^(1/2))^(1/2)(2)

The solution follows similarly as above from the observation that if typeset structure denotes the right hand side of  (2) then typeset structure satisfies the functional equation typeset structure which solution is typeset structure.  

The original Ramanujan’s proof is incomplete. The gaps were filled by Vijayaraghavan  [3, p.348]  and Herschfeld  [4] .  The result can also be found in  [5] .

For general infinite nested radicals see  

References

[1]  Ramanujan, S. (1911). Question No. 298. Journal of the Indian Mathematical Society, III, 90.

[2]  Ward, A. (1996). Problem 80.E. The Mathematical Gazette, July, 422.

[3]  Ramanujan, S. (1927). Collected papers. Edited by G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson.. Cambridge: University Press.

[4]  Herschfeld, A. (1935). On infinite radicals. Amer. Math. Monthly , 42, 419-429.

[5]  Alexanderson, G. L., Klosinski, L. F., & Larson, L. C. (1985). The William Lowell Putnam Mathematical Competition. Problems and solutions: 1965-1984. Washington, D. C.: The Mathematical Association of America. Distr. by John Wiley & Sons. .

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Štefan Porubský: 3.

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