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