A limit of polynomials as a $q$-series

Fix a positive integer $ k$ . Let $ b_n(q)$ be the polynomial defined by the recursive equation $ $ b_n(q)=\binom{n+k-1}k+(1+q^{n-1})b_{n-1}(q), \qquad n\geq1,$ $ initializing with $ b_0(q)=0$ .

I run into the below through experiment.

QUESTION. Denote $ (q)_m=(1-q)(1-q^2)\cdots(1-q^m)$ . Is this true? $ $ \lim_{n\rightarrow\infty}\left(b_n(q)-\binom{n+k}{k+1}\right) =\sum_{m=1}^{\infty}\frac{q^{\binom{m+1}2}}{(q)_m(1-q^m)^{k+1}}.$ $