20
$\begingroup$

How would one go about computing the value of $X$, where

$X=5^0+ \sqrt{5^1+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+\sqrt{5^{16}+\sqrt{5^{32}+\dots}}}}}}$

I have tried the standard way of squaring then trying some trick, but nothing is working. I have also looked at some nested radical previous results, but none seem to be of the variety of this problem. Can anyone come up with the answer? Thank

1 Answers 1

21

The trick is to pull out a $\sqrt{5}$ factor from the second term:
$$ \frac{\sqrt{5^1+ \sqrt{5^2 + \sqrt{5^4 + \sqrt{5^8 + \cdots}}}}}{\sqrt{5}} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}, $$ which I call $Y$ for convenience. To see why this is true, observe that $$ \begin{align*} \frac{\sqrt{5^1+x}}{\sqrt{5}} = \sqrt{1+\frac{x}{5^1}} \\ \frac{\sqrt{5^2+x}}{5^1} = \sqrt{1+\frac{x}{5^2}} \\ \frac{\sqrt{5^4+x}}{5^2} = \sqrt{1+\frac{x}{5^4}} \end{align*} $$ and so on. Applying these repeatedly inside the nested radicals gives the claim. (The wording of this explanation has been inspired mostly by a comment of @J.M. below.)

Now, it only remains to compute $Y$ and $X$. By squaring, we get $$ Y^2 = 1 + Y \ \ \ \ \implies Y = \frac{\sqrt{5}+1}{2}, $$ discarding the negative root. Plugging this value in the definition of $X$, we get: $$ X = 1 + \sqrt{5} Y = 1 + \frac{5+\sqrt{5}}{2} = \frac{7+\sqrt{5}}{2} . $$

Note on the convergence issues. As @GEdgar points out, to complete the proof, I also need to demonstrate that both sides of the first equation do converge to some finite limits. For our expressions, convergence follows from @Bill Dubuque's answer to my question on the defining the convergence of such an expression. I believe that with some work, one can also give a direct proof by showing that this sequence is bounded from above (which I hope will also end up showing the theorem Bill quotes), but I will not pursue this further. Added: See @Aryabhata's answer to a related question for a hands-on proof.

  • 4
    @Fool: you mean the nested radical? Note that $\sqrt{5+x}/\sqrt 5=\sqrt{1+x/5}$; just keep applying that as you go deeper.2011-08-31
  • 0
    @J.M. I thought the expressions involved in my original explanation of the trick was scary-looking, and preferred your explanation. Is it ok if I rewrite my answer with this style? (I'll add a note acknowledging you in a moment.)2011-08-31
  • 2
    Technically, you need to prove convergence before you can do that.2011-08-31
  • 0
    @GEdgar Good point. Can you point out exactly which step needs further justification?2011-08-31
  • 0
    No problem with that, Srivatsan. @GEdgar: Correct, but since this is tagged [tag:algebra-precalculus]... ;)2011-08-31
  • 1
    AHA! precalculus mean we don't need no stinkin' proofs ??2011-08-31
  • 1
    $Y^2=1+Y$ has $Y=\infty$ as solution, as well as the one you found...2011-08-31
  • 0
    @GEdgar: Heh. I'm trying to think of something that avoids invoking Banach (the likelihood of that being discussed before a calculus course is, well...) but I'm drawing a blank. Got ideas?2011-08-31
  • 1
    Related: http://math.stackexchange.com/questions/11945/limit-of-sqrt7-sqrt7-sqrt7-cdots/11969#119692011-09-01
  • 0
    @Aryabhata: That's it! (I couldn't find the question since it didn't have the search terms I was using). Thanks!2011-09-01