  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 negligable. The
anomalous dimensions of the leading twist operators are obtained in the
next-to-leading order.
