International Mathematical Olympiad 1968 Problem 6

For every natural number $n$, evaluate the sum $$\sum_{k=0}^{\infty} \left[ \frac{n + 2^k}{2^{k+1}} \right] = \left[ \frac{n + 1}{2} \right] + \left[ \frac{n + 2}{4} \right] + \cdots + \left[ \frac{n + 2^k}{2^{k+1}} \right] + \cdots$$

(The symbol $[x]$ denotes the greatest integer not exceeding $x$.)