  In this paper, we study the asymptotic behavior of the following function
$$M_k(n):=(-1)^{k-1}
\sum_{j=0}^{k-1}\big(p(n-j(3j+1)/2)-p(n-j(3j+5)/2-1)\big),$$ which arises from
Andrews and Merca's truncated pentagonal number theorem.
