evaluate this.

Well the problems seems to be difficult. I haven't been able so far to solve it entirely but this is what I have done. First thing to note is that the given sum exists. We consider a generalization:
f(x) = \sqrt{x+\sqrt{(x+1)+\sqrt{(x+2)+\ldots}}}
so that we are trying to find f(1). Its easy to see that f(x) satisfies a non-linear functional equation
f(x)^2 = x + f(x+1)
If some how we could solve this equation for f(x), then we can find the given sum.
Usually such kind of nested infinite radicals are solved this way. A famous sum of a similar kind was given by Ramanujan. He asked for the sum of the infinite nested radical
\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\ldots}}}}
(This equals 3)

p.s.: could you quote the source of the problem?

Can you prove that the sum exists ?

I am looking for the proof :-)

http://pballew.net/1935Herschfeld.pdf

Go to the above link and you will find what you need , Sir .

That's really interesting. Could you give us some reference for this. I am unable to find any google results for this keyword

This is a constant named as " Kasner Costant " . Its exact closed form expression hasn ' t been found out yet , but its approximate value , is 1 . 7533 .

sir, our 1st sem maths teacher gave this problem to us. however he neither told the method of evaluating this nor the answer..

the actual question was... 'whether the the sum existed and if it did, whats the answer??'
bt i don't know the actual source of this problem.. :(

I hope you are talking about this:
\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\ldots}}}}

@ rishabh, how can you approximate √1+√2+√3+...∞ = √2+√3+...∞ = x ??
we could have done that for √1+√1+√1+.....∞

so, i don't think it is right..

i considered the answer to be some x.
then as this is an infinite series then we can approximate √1+x = x
=> on suaring x =1+√52

@fahad then 1 + 1/2 + 1/3 ................... should also not have any algebraic value but we still write it as 1/ (1-1/2) = 2

√π