# An Infinite Continued Fraction

Calculus Level 5

$\Large 1+\frac{2}{1+\frac{2}{ 1+\frac{2}{\ddots+\frac{2}{x}}}}=2$

The above infinite continued fraction can be viewed as a recurrence relation: $$a_0=x$$, $$a_{k+1}=1+\frac{2}{a_{k}}$$ for $$k\ge 0$$ and $$\displaystyle\lim_{k \to \infty}a_k=2$$.

What is the range of integers $$x$$ that satisfy this condition?

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