  We develop an algebraic approach to the analytic bootstrap in CFTs. By acting
with the Casimir operator on the crossing equation we map the problem of doing
large spin sums to any desired order to the problem of solving a set of
recursion relations. We compute corrections to the anomalous dimension of large
spin operators due to the exchange of a primary and its descendants in the
crossed channel and show that this leads to a Borel-summable expansion. We
analyse higher order corrections to the microscopic CFT data in the direct
channel and its matching to infinite towers of operators in the crossed
channel. We apply this method to the critical $O(N)$ model. At large $N$ we
reproduce the first few terms in the large spin expansion of the known two-loop
anomalous dimensions of higher spin currents in the traceless symmetric
representation of $O(N)$ and make further predictions. At small $N$ we present
the results for the truncated large spin expansion series of anomalous
dimensions of higher spin currents.
