In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.