Why can't one solve this continued fraction this way?

I have the following problem:

The way I solved the problem is by rewriting it as a system of equations:

$$x = 1 +\frac 1y$$ $$y = 2 + \frac1y$$ Then I solved it as a normal system of equations and arrived at the answer of $$\sqrt2$$. But apparently the answer is $$e$$? How does that make sense? Can someone explain the result to me?

• Can you show us how you solved it? – Randall Jul 22 at 18:17
• This is equal to $\sqrt{2}$. I don't know where you got this equals $e$ from. – Peter Foreman Jul 22 at 18:22
• Continued fraction for $e$ is not periodic – J. W. Tanner Jul 22 at 18:28
• @Adam Grey:$\;$Out of curiosity, what book? – quasi Jul 22 at 18:28
• @Adam Grey: To solve without a system, consider the continued fraction for $x+1$. Then immediately, you get the equation $$x+1=2+\frac{1}{x+1}$$ which yields $x=\sqrt{2}$. – quasi Jul 22 at 18:33

As far as I know, $$e$$ is not so good looking at continued fractions... Indeed $$e=[2;1,2,1,1,4,1,1,6,1,1,8,\ldots]$$You have found a correct answer, yes! $$x=\sqrt{2}$$ is the correct number with the given continued fraction.
• Do you know that \ldots and \dots are rendered in the same way. – Peter Foreman Jul 22 at 19:20