  Asymptotic study on the partition function $p(n)$ began with the work of
Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$
and an error bound was given by Lehmer. Despite having this, a full asymptotic
expansion for $p(n)$ with an explicit error bound is not known. Recently
O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th
powers, initiated by Wright, and consequently obtained an asymptotic expansion
for $p(n)$ along with a concise description of the coefficients involved in the
expansion but without any estimation of the error term. Here we consider a
detailed and comprehensive analysis on an estimation of the error term obtained
by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$.
This gives rise to an infinite family of inequalities for $p(n)$ which finally
answers to a question proposed by Chen. Our error term estimation predominantly
relies on applications of algorithmic methods from symbolic summation.
