Alternating series of compositions of triangular numbers I'm modeling a process which involves a subset $S$ of a large number $n_A$ of objects - call them balls.  Each time I add a ball to $S$, it may dislodge another ball with probability proportional to the fraction of the total balls already in $S$.
I've gotten as far as this expression for the expected number of balls $n_k$ after $k$ additions, and I'm hoping someone might show me how to find a closed form expression for this series, or at least some bounds:
$$\langle n_{k}\rangle=k-
\frac{1}{n_{A}}\sum_{a=1}^{k-1}1+
\frac{1}{{n_{A}}^2}\sum_{a=1}^{k-2}\sum_{b=1}^{a}1-
\frac{1}{{n_{A}}^3}\sum_{a=1}^{k-3}\sum_{b=1}^{a}\sum_{c=1}^{b}1+
\frac{1}{{n_{A}}^4}\sum_{a=1}^{k-4}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}1-...$$

Note that
$$\sum_{a=1}^{k-4}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}1=
\sum_{a=1}^{k-4}\sum_{b=1}^{a}\sum_{c=1}^{b}\binom c1=\sum_{a=1}^{k-4}\sum_{b=1}^{a}\binom {b+1}2=\sum_{a=1}^{k-4}\binom {a+2}3=\binom {k-1}4$$
and so on.
Hence
$$\begin{align}
\langle n_k\rangle
&=k-
\frac{1}{n_{A}}\sum_{a=1}^{k-1}1+
\frac{1}{{n_{A}}^2}\sum_{a=1}^{k-2}\sum_{b=1}^{a}1-
\frac{1}{{n_{A}}^3}\sum_{a=1}^{k-3}\sum_{b=1}^{a}\sum_{c=1}^{b}1+
\frac{1}{{n_{A}}^4}\sum_{a=1}^{k-4}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}1-...\\
& =k-\frac 1{n_A}\binom {k-1}1+\frac 1{n_A^2}\binom {k-1}2-\frac 1{n_A^3}\binom{k-1}3+\frac 1{n_A^4}\binom {k-1}4-\cdots\\
&=\sum_{r=0}^\infty  \binom {k-1}r\left(- \frac 1{n_A}\right)^r\\
&=\sum_{r=0}^{k-1}  \binom {k-1}r\left(- \frac 1{n_A}\right)^r
\qquad\qquad\text{as } \binom {k-1}r=0 \text {  for }r>k-1\\
&=\left(1-\frac1{n_A} \right)^{k-1}\qquad\blacksquare
\end{align}$$
