Ramanujan would like it

Algebra Level 3

1 + 2 1 + 3 1 + 4 1 + = ? \sqrt{1 + 2\sqrt{1 + 3\sqrt{1+ 4\sqrt{1+ \cdots}}}} =\, ?


The answer is 3.

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1 solution

Chew-Seong Cheong
Feb 24, 2016

Ramanujan discovered that:

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 ) . . . \small x+n+a = \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n)\sqrt{a(x+2n) + (n+a)^2 + (x+2n)\sqrt{...}}}}

When x = 2 x = 2 , n = 1 n=1 and a = 0 a=0 , then we have:

2 + 1 + 0 = 1 + 2 1 + 3 1 + 4 1 + . . . = 3 2+1+0 = \sqrt{1+ 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1+...}}}} = \boxed{3}

yes that is great ramanujan's generalization.

Sathvik Acharya - 4 years, 3 months ago

You are absolutely correct my friend

Hrishikesh Harish - 5 years, 3 months ago

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