  Various methods to obtain the analytic continuation near $z=1$ of the
hypergeometric series $_{p+1}F_p(z)$ are reviewed together with some of the
results. One approach is to establish a recurrence relation with respect to $p$
and then, after its repeated use, to resort to the well-known properties of the
Gaussian hypergeometric series. Another approach is based on the properties of
the underlying generalized hypergeometric differential equation: For the
coefficients in the power series expansion around $z=1$ a general formula,
valid for any $p$, is found in terms of a limit of a terminating
Saalsch\"utzian hypergeometric series of unit argument. Final results may then
be obtained for each particular $p$ after application of an appropriate
transformation formula of the Saalsch\"utzian hypergeometric series. The
behaviour at unit argument of zero-balanced hypergeometric series, which have
received particular attention in recent years, is discussed in more detail. The
related problem involving the behaviour of partial sums of such series is
addressed briefly.
