Andrews and Merca investigated a truncated version of Eulers pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobis 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 Eulers pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobis triple product identity in a different manner.
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