Two truncated identities of Gauss

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.