How can I find the limiting value of this expression:
$$\sum_{j=1}^N \mu_j^2 \sum_{n_j = 1}^\infty \frac{n_j^2 \overline{n}_j^{n_j}}{(1 + \overline{n}_j)^{1 + n_j}},$$
where
$$\mu_j = \left(j - \frac{N + 1}{2}\right)$$
and $\overline{n}_j$ is some average number. I would like to evaluate the above in terms of some specificed $\overline{n}_j \in Z^+$. Note $\overline{n}_j \neq \overline{n_j}$.
\begin{eqnarray*} \sum_{n=1}^{\infty} n^2x^{n} =\frac{x(1+x)}{(1-x)^{3} } \end{eqnarray*} Let $\lambda= \bar{n_j}$ Each inner sum can written as \begin{eqnarray*} \sum_{n=1}^{\infty} \frac{n^2\lambda^{n}}{(1+\lambda)^{1+n} } =\frac{1}{1+\lambda} \frac{\frac{\lambda}{1+\lambda}\left(1+\frac{\lambda}{1+\lambda}\right)}{(1-\frac{\lambda}{1+\lambda})^3}=\lambda(1+2\lambda) \end{eqnarray*} So the original sum can be written as \begin{eqnarray*} \sum_{j=1}^{N}\mu_j^{2} \bar{n}_j (1+2 \bar{n}_j) \end{eqnarray*} Without further knowledge of the $ \bar{n}_j$ 's it is not possible to proceed from here.