Truncated sums for the partition function and a problem of Merca

In 2012, Andrews and Merca established a truncated form of Euler’s pentagonal number theory. In this note, using the Rogers–Fine identity, we present a partition theoretic interpretation of truncated sums on the partition function which solve a problem given by Merca. Furthermore, we prove an inequality on ordinary partition which implies the positive result on truncated sums of partition function due to Andrews and Merca.