Grasshopper on the Number Line

A grasshopper sits at $0$ on the integer number line and repeatedly jumps to the right, each jump covering $1$ unit or $2$ units with probability $\tfrac{1}{2}$ each, independently forever. For a positive integer $k$, let $p_k$ denote the probability that the grasshopper ever touches down exactly on $k$. Which $k > 0$ maximizes $p_k$, and what is that maximum probability?