Markov Chains & Long-Run Behavior · Superchallenge problem

A walker on the four vertices of a tetrahedron moves each step to one of the other three vertices, chosen uniformly. Let $p_n$ be the probability of being at the starting vertex after $n$ steps ($p_0 = 1$). Evaluate $\displaystyle\sum_{n=0}^{\infty}\left(p_n - \tfrac14\right)$. Express your answer as a common fraction.