Infinite sqrt Series?

  • Thread starter PEZenfuego
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  • #1
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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:

[tex]\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5...}}}}}[/tex]

Any thoughts would be very much appreciated.
 

Answers and Replies

  • #2
Char. Limit
Gold Member
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Taken from Wolfram Mathworld:

Wolfram Mathworld said:
Herschfeld (1935) proved that a nested radical of real nonnegative terms converges iff [tex]x_n^{2^{-n}}[/tex] is bounded.

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

[tex]f(n)=n^{2^{-n}}[/tex]

And we find that indeed, this function is bounded. So yes, your nested radical is convergent.
 
  • #3
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[tex]\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt{5...}}}}}\approx 1.7579327566180045327[/tex]
 

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