# Simplify continued fraction with $\pi$

I'm not even sure where to start on this:

Simplify: $$\pi+\dfrac{2}{\pi+\dfrac{2}{\pi +\dfrac{2}{\dots}}}$$

The second term is a rational expression with 2 in the numerator and the denominator is the entire expression again...over and over with no end.

One thought I had was to let $x=\pi + 2/x$, then multiply both sides by x to get a quadratic. However, I have turned an expression into an equation, so this doesn't seem right.

$x^2-\pi x-2=0$ and solve using q.e. This would give $x = (\pi \pm \sqrt{\pi^2+8} )/2$ However, the problem said to simplify not solve.

Any thoughts?

• Your thought was right. – André Nicolas Jul 12 '14 at 19:41
• Note, for the $\pm$, can it really be minus? Hint: $\sqrt{\pi^2+8}>\pi$... – Thomas Andrews Jul 12 '14 at 19:48
• You are given an expression. You write an equation involving the expression. You solve the equation to get the value of the expression. Nothing wrong with that! – TonyK Jul 12 '14 at 20:06
• Thank you very much. This helped tremendously! Yes, I can see that it can't be the minus part of the Q.E. – user163862 Jul 12 '14 at 20:30

## 1 Answer

As Thomas pointed out,

This expression is clearly pozitive so the answer is $\dfrac{(\pi + \sqrt{\pi^2+8} )}2$.