Help Infinite continued fractions

[FP3]

Help! Infinite continued fractions!

Homework Statement

-2 + 1
-2 + 1
-2 + 1
-2 + 1
...

an = -2, bn = 1

Homework Equations

What is the sum of this indefinite continued fraction, putting this into a quadratic equation? x^2 + 2x - 1 = 0

The Attempt at a Solution

Using the quadratic formula, I got .4142 and -2.4142, but there is only 1 sum. The an value is -2 (so is that even possible?). Help quickly!

 

[Dick]

It's going to be the negative root, isn't it? Write down a few more approximations, like -2+1/-2 and -2+1/(-2+1/(-2)) to make it clear.

 

[KevB]

From the .doc image:

x = [-2, -2, -2,...] = [-2],[0] = [-3, 1, 1, 2],[0] =$$-1 - \sqrt{}2$$
~ -2.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727

 

[SammyS]

$$\text{Let }x=-2+\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\dots}}}}\quad\to\quad x+2=\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\dots}}}}\quad\to\quad \frac{1}{x+2}=-2+\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\frac{1}{-2+\dots}}}}$$

But the right hand side of the last expression is x.

Therefore, x = 1/(x+2).

Solve that for x.