  If the step distribution in a renewal process has finite mean and regularly
varying tail with index -{\alpha}, 1<{\alpha}<2, the first two terms in the
asymptotic expansion of the renewal function have been known for many years.
Here we show that, without making any additional assumptions, it is possible to
give, in all cases except for {\alpha}=3/2 , the exact asymptotic behaviour of
the next term. In the case {\alpha}=3/2 the result is exact to within a slowly
varying correction. Similar results are shown to hold in the random walk case.
