  In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal
number theorem and showed the nonnegativity related to partition functions.
Meanwhile, Andrews-Merca and Guo-Zeng independently conjectured that the
truncated Jacobi triple product series has nonnegative coefficients, which has
been confirmed analytically and also combinatorially. In 2022, Merca proposed a
stronger version for this conjecture. In this paper, by applying Agarwal,
Andrews and Bressoud's Bailey lattice, we derive a truncated version for the
Jacobi triple product series with odd basis which reduces to the Andrews-Gordon
identity as a special instance. As consequences, we obtain new truncated forms
for Euler's pentagonal number theorem, Gauss'theta series on triangular numbers
and square numbers, which lead to inequalities for certain partition functions.
Moreover, by considering a truncated theta series involving $\ell$-regular
partitions, we confirm a conjecture proposed by Ballantine and Merca about
6-regular partitions and show that Merca's stronger conjecture on truncated
Jacobi triple product series holds when $R = 3S$ for $S \geq 1.$
