distinct parts and partitions with less than k parts

Statement

\prod_{n \geq 1} (1 + x q^{n}) = \sum_{k \geq 0} \frac{q^{k (k + 1) / 2} x^{k}}{(1 - q) (1 - q^{2}) \dots (q - q^{k})}

Note that the LHS is the generating function for partitions with distinct parts and the denominator for the RHS is the generating function for partitions with less than or equal to $k$ parts (by this)

Proof

We construct a bijection between the two as counting number of partitions. For a partition with distinct parts, shift it, cut off the staircase, and transpose. This yields a bijection.