Euler pentagonal identity (${\prod }_{n\ge 1}(1-q^n)={\sum }_{r\in \mathbb{Z}}(-1{)}^r{q}^{r(3r+1)/2}$)

The identity holds for $|q|<1$, with both sides locally uniformly convergent. In the character theta proof of the eta product, pair the terms indexed by $n$ and $-n$ and then select $n=6r+1$; their Dirichlet character signs are $(-1{)}^r$. Removing $q^{1/24}$ gives the displayed product identity.