Is the Infinite Sqrt Series Finite?

[PEZenfuego]

I heard something interesting today, but I am skeptical. I heard that a certain infinite series has a finite answer and I was curious as to whether or not this was true and if we can prove it either way.

This is the series:

$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5...}}}}}$$

Any thoughts would be very much appreciated.

 

[Char. Limit]

Taken from Wolfram Mathworld:

Herschfeld (1935) proved that a nested radical of real nonnegative terms converges iff $$x_n^{2^{-n}}$$ is bounded.

So, the question is, is x_n^(2^(-n)) bounded? To answer this, we realize that the n^th term of x is simply n, and so we check the function...

$$f(n)=n^{2^{-n}}$$

And we find that indeed, this function is bounded. So yes, your nested radical is convergent.

 

[Xitami]

$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5...}}}}}\approx 1.7579327566180045327$$