A positive number $x$ satisfies $x=3+1$/(3 + 1/(3 + …)), the pattern continuing without end. Then $x$ equals:

The whole tail after the first 3 repeats the original expression, so $x=3+1/x$. Multiplying by $x$ gives $x^2-3x-1=0$, so $x$ = (3 ± √13)/2, and only the positive sign can equal an expression bigger than 3. (The negative root (3 − √13)/2 ≈ −0.30 is rejected; (√13 − 3)/2 solves $x^2+3x-1=0$ instead; (3 + √5)/2 belongs to $x=1+1/x$.) Numerically $x$ ≈ 3.3028, and 3 + 1/3.3028 ≈ 3.3028.