Puzzle: one over one plus … – methods of solution

April 2, 2007 pm30 6:59 pm

This is the place to offer methods of solutions for the puzzle below:

What is the value of $\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$ ?

A month ago Dave Marain ran a discussion of $\sqrt{2 + \sqrt{2 + \sqrt{2 + \dotsb}}}$ over here.

I know this one has been done many many times before. But he generated a nice discussion.

There are also places for:

What makes these two problems so similar?

5 Comments leave one →

Edgardo permalink

April 3, 2007 pm30 9:48 pm 9:48 pm

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \dotsb}}}$

First examine what this ‘object’ is. It looks a like a sequence,
or better, the limit of the sequence.
The sequence can be described by:

$a_1 = b \hspace{1cm}\mathrm{(initial value)}$
$a_2 = \sqrt{2+a_1}$
$a_3 = \sqrt{2+a_2} = \sqrt{2+ \sqrt{2+a_1}}$

In general:
$a_{n+1}= \sqrt{2+ a_{n}}$

If the limit exists, then $a_{n}$ and $a_{n+1}$ have the same limit.
Thus, $a_{n}=a_{n+1}=B$ , where $B$ is the limit of the sequence.

Pluggin this into the equation
$a_{n+1}= \sqrt{2+ a_{n}}$
yields: $B = \sqrt{2 + B}$
This leads to a quadratic equation:
$B^2-B-2=0$

Solving for B yields (B>0):
$B = 2$.

Of course, one has to prove the several steps.

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Edgardo permalink

April 3, 2007 pm30 10:00 pm 10:00 pm

Correction for my post above:

If the limit exists, then $a_{n}$ and $a_{n+1}$ have the same limit. This can be written as
$\rm{lim} a_{n} = B$ and $\rm{lim} a_{n+1} = B$
I forgot to write $\rm{lim}$ in my former post.

———–

Now, I think I know how this sequence was generated:

$2 = \sqrt{2+2}$
Plugging this into the right hand side yields:
$2 = \sqrt{2+ \sqrt{2+2}}$
and so on…

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Edgardo permalink

April 3, 2007 pm30 10:20 pm 10:20 pm

I just figured out how such a sequence looks for 3 (this is fun):

$3 = \sqrt{9}= \sqrt{6+3}$
$= \sqrt{ 6+ \sqrt{6+3} }$
$= \sqrt{ 6+ \sqrt{ 6 + \sqrt{ 6 + 3 }}}$
$ latex = \sqrt{ 6 + \sqrt{ 6 + \sqrt{ 6 + \dotsb}}}$

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Edgardo permalink

April 3, 2007 pm30 11:17 pm 11:17 pm

$\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$

This can be analyzed in a similar way.
The ‘object’ is the limit of a sequence:

$a_{1} = b (\mathrm{initial value})$
$a_{2} = 1+ \frac{1}{a_{1}}$
$a_{3} = 1 + \frac{1}{a_{2}} = 1 + \frac{1}{1+ \frac{1}{a_{1}}}$

In general:
$a_{n+1} = 1 + \frac{1}{a_{n}}$

If we assume that the sequence $a_{n}$ converges, then
$a_{n}$ and $a_{n+1}$ have the same limit. This yields

$B = 1 + \frac{1}{B}$ , where B is the limit of the sequence.
It follows:
$B^2-B-1 = 0$
Solving for B yields (B>0):

$B = \frac{1}{2}(\sqrt{5}+1)$

How was this constructed?

$\frac{1}{2} \left(\sqrt{5}+1 \right) = \frac{1}{2} \left(\sqrt{5}+1 \right) + 1 - 1$
$= 1+ \frac{1}{2} \left(\sqrt{5}+1 \right) -1$
$= 1+ \frac{\sqrt{5}-1}{2}$
$= 1+ \frac{5-1}{2(\sqrt{5}+1)}$
$= 1+ \frac{4}{2(\sqrt{5}+1)}$
$= 1+ \frac{2}{\sqrt{5}+1}$
$= 1+ \frac{1}{\frac{1}{2}(\sqrt{5}+1)}$

The tactic is obvious, in the first step a 1 is added because we want to have the 1 + 1/(1+/…) term. The remaining task is to manipulate the term after the 1 in such a way, that
$\frac{1}{2} \left(\sqrt{5}+1 \right)$

appears again.

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Edgardo permalink

April 3, 2007 pm30 11:26 pm 11:26 pm

I just noticed that I calculated the value of
$1+\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$
instead of
$\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$

Thus, the limit of
$\frac{1}{1+\frac{1}{1 + \frac{1}{1 + \dotsb}}}$
is
$\frac{1}{2}(\sqrt{5}-1)$

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