  We have calculated the contributions proportional to
$\beta_0^n\alpha_s^{n+1}$ to the longitudinal fragmentation function in
$e^+e^-$ annihilation to all orders of perturbation theory. We use this result
to estimate higher-order perturbative corrections and nonperturbative
corrections to the longitudinal cross section $\sigma_L$ and discuss the
prospects of determining $\alpha_s$ from $\sigma_L$. The structure of infrared
renormalons in the perturbative expansion suggests that the longitudinal cross
section for hadron production with fixed momentum fraction $x$ receives
nonperturbative contributions of order $1/(x^2Q^2)$, whereas the total cross
section has a larger, $1/Q$ correction. This correction arises from very large
longitudinal distances and is related to the behaviour of the Borel integral
for the cross section with fixed $x$ at large values of the Borel parameter.
