IR-Renormalon Contributions to the Structure Functions $g_3$ and $g_5$

We calculate the leading $1/N_f$ perturbative contributions to the polarized nonsinglet structure functions $g_3$ and $g_5$ to all orders in $\alpha_s$. The contributions from the first renormalon pole are determined. It is a measure for the ambiguity of the perturbative calculation and is assumed to dominate the power corrections. The corrections $\Delta g_3$ and $\Delta g_5$ are given as functions of the Bjorken variable $x$ and turn out to be negligible. The anomalous dimensions of the leading twist operators are obtained in the next-to-leading order