Jacobi Identity guide

Question

Can any one guide me how can I prove this identity. $$\prod_{n=1}^{\infty}(1-q^j)^3=\sum_{n=0}^{\infty}(-1)^n(2n+1)q^{(n^2+n)/2}$$

Answer 1

A proof of this formula, $$ \prod_{i\ge 1}(1-q^i)^3=\sum_{j=0}^{\infty}(-1)^j(2j+1)q^{\frac{j(j+1)}{2}} $$ due to Jacobi, can be found in the paper of Josuat-Verges and Kim, section $10$, page $26$, $(56)$. It is derived from Theorem $9.1$, on page $24$.