# 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. Jul 12 '14 at 19:41
• Note, for the $\pm$, can it really be minus? Hint: $\sqrt{\pi^2+8}>\pi$... 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! 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. Jul 12 '14 at 20:30

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