  Andrews and Merca investigated a truncated version of Euler's pentagonal
number theorem and showed that the coefficients of the truncated series are
nonnegative. They also considered the truncated series arising from Jacobi's
triple product identity, and they that its coefficients are nonnegative. This
conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee
using different approaches. In this paper, we provide a new combinatorial proof
of their nonnegativity result related to Euler's pentagonal number theorem.
Meanwhile, we find an analogous result for a truncated series arising from
Jacobi's triple product identity in a different manner.
