Computing Multi-Step Improper Integrals

Define a function $f$ as follows: For every integer $k$, $f(k) = 0$. For all $x \in (k, k+1)$, $$f(x) = \begin{cases} \frac{2^{-k}}{\sqrt{(x-k)}} & \text{if }k\text{ is even} \\ \frac{2^{-k}}{\sqrt{-(x-k-1)}} & \text{if }k\text{ is odd} \end{cases}$$ What is the value of $\int_0^\infty f(x) \, \text{d}x$?

(Hint: You're going to want to split the larger integral up into infinitely many smaller integrals.)