The value of $$4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cfrac{1}{4 + \ldots \infty}}}}$$ is:

Let the value of the infinite continued fraction be $$x = 4 + \cfrac{1}{5 + \cfrac{1}{4 + \cfrac{1}{5 + \cdots}}}$$. Since the pattern repeats with alternating 4 and 5, we can write $$x = 4 + \frac{1}{5 + \frac{1}{x}}$$ because after one full cycle of (4, 5), the continued fraction returns to its original form.

Simplifying the inner fraction: $$5 + \frac{1}{x} = \frac{5x + 1}{x}$$, so $$x = 4 + \frac{x}{5x + 1} = \frac{4(5x + 1) + x}{5x + 1} = \frac{20x + 4 + x}{5x + 1} = \frac{21x + 4}{5x + 1}$$.

Cross-multiplying: $$x(5x + 1) = 21x + 4$$, which gives $$5x^2 + x = 21x + 4$$, or $$5x^2 - 20x - 4 = 0$$. Using the quadratic formula: $$x = \frac{20 \pm \sqrt{400 + 80}}{10} = \frac{20 \pm \sqrt{480}}{10}$$.

Now $$\sqrt{480} = \sqrt{16 \times 30} = 4\sqrt{30}$$, so $$x = \frac{20 \pm 4\sqrt{30}}{10} = 2 \pm \frac{2\sqrt{30}}{5}$$. Since the continued fraction is positive and greater than 4, we take the positive root: $$x = 2 + \frac{2\sqrt{30}}{5} = 2 + \frac{2}{5}\sqrt{30}$$.

This matches option (1).