Ramanujan's nested radicals

Calculus Level 2

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

∞ \infty 3 3 9 9 2 2

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Atul Kumar Ashish by Atul Kumar Ashish , 21, India

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3 solutions

( x + a ) = x 2 + a 2 + 2 a x = a 2 + x ( x + 2 a ) = a 2 + x a 2 + ( x + a ) ( x + 3 a ) = a 2 + x a 2 + ( x + a ) a 2 + ( x + 2 a ) ( x + 4 a ) = a 2 + x a 2 + ( x + a ) a 2 + ( x + 2 a ) a 2 + ( x + 3 a ) . . . Now setting a=1, x=2 we get the Ramanujan’s Nested Radical (x+a) \\ =\sqrt{x^2+a^2+2ax} \\ =\sqrt{a^2+x(x+2a)} \\ =\sqrt{a^2+x\sqrt{a^2+(x+a)(x+3a)}} \\ =\sqrt{a^2+x\sqrt{a^2+(x+a)\sqrt{a^2+(x+2a)(x+4a)}}} \\ =\sqrt{a^2+x\sqrt{a^2+(x+a)\sqrt{a^2+(x+2a)\sqrt{a^2+(x+3a)\sqrt{...}}}}} \\ \text{Now setting a=1, x=2 we get the Ramanujan's Nested Radical} Try out these crazy nested radicals!

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* Nested radical

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Ku John
Aug 6, 2016

pls i arld posted this weeks ago

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