On the positivity of truncated pentagonal number series and some conjectures of Merca

Let $ν_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+ν_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};