Consider a sequence of functions as follows:-

$ f_1 (x) = \sqrt {1+\sqrt {x} } $

$ f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } } $

$ f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } } $

……and so on to

$ f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } } $

**Evaluate this function as n tends to infinity.**

Or logically:

Find

$ \displaystyle{\lim_{n \to \infty}} f_n (x) $ .

### Solution

Ramanujan discovered

$$ x+n+a=\sqrt{ax + (n+a)^2 +x \sqrt{a(x+n)+(n+a)^2 +(x+n) \sqrt{\ldots}}} $$

which gives the special cases

$$ x+1=\sqrt{1+x \sqrt{1 + (x+1) \sqrt{1 + (x+2) \sqrt{1 + (x+2) \sqrt{\ldots}}}}}$$

for *x=2 , n=1* and *a=0*

$$3= \sqrt{1+2 \sqrt{1+3 \sqrt{1+ 4 \sqrt{1+\cdots}}}}$$

Comparing these two expressions & assuming

=$ X $ , we can write the problem as:

$ \displaystyle {\lim_{n \to \infty}} f_n (x) $

= $ \sqrt {1+X} $

= $ \sqrt {1+3} $

=$ \sqrt {4} $

=$ 2 $

–

**For further info please refer the comments below. There is also a supportive article on Ramanujan Nested Radicals on this blog.**

Carl Andreas

I’m not entirely sure, but using C++ (with n = 1,000,000) I numerically evaluated it to the function f(x) = 2. But as I said, not quite sure!

Gaurav Happy Tiwari

You’re right..! After using google, I got this Link , which was also saying the same. But I wasn’t satisfied.

animesh

Answer is 3 according to formulation 27 at http://mathworld.wolfram.com/NestedRadical.html .

Dave Radcliffe

There are two slightly different versions of this nested radical, so you need to be careful.

The version posed by Ramanujan was

sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + … = 3

Your version is almost the same:

sqrt(1 + 1*sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + … = sqrt(1 + 3) = 2.

Tagus from Reddit

Hi there,

Just a small typo — I think you meant to write the limit as n tends to infinity. On all of the limits you wrote in that article, you unfortunately said that x goes to infinity.

x_x

Gaurav

x_x Corrected Now. Thanks.

tinkerbelle86

wow – wouldnt have a clue where to start!

Udit Agarwal

I think you might be one of the best bloggers in India today. We are having a TEDx conference, and it would be great to have you as a Speaker. I am sure you can come up with a very interesting talk. Let me know however I can contact you.

karthik

Ramanujan always the best

Ele Munjeli (@elemunjeli)

I really love the version that starts 3= because it smells like a magic number but really implicates the architecture of the number system we use.

narendra hegade

really nice.

Victor Kamat

Ramanujan isn’t human, he is a Beast

Gaurav Tiwari

You stole my words! I totally agree with you.

lankesh

second equation is wrong x+1=sq(1+xsq(1+(x+1))sq(1+(x+2)sq(1+(x+3))))

kolomedukasi.com

thank you , this is one of my try out question in my school.

quiet confused since i saw this crazy square root .lol