\begin{equation*} \begin{aligned} P(\text{1st success on an odd trial}) \amp = f(1) + f(3) + f(5) + ...\\ \amp = p + (1-p)^2 \cdot p + (1-p)^4 \cdot p + (1-p)^6 \cdot p + ... \\ \amp = p \sum_{x=0}^{\infty} \left ( (1-p)^2 \right)^k \\ \amp = p \frac{1}{1-(1-p)^2} \approx 0.5681 \end{aligned} \end{equation*}