The given sum is:

$S = \sum_{n=1}^{\infty} \frac{n^2 e^{-n}}{(n+1)^3}$

To simplify or analyze this series, let's take the following approach:

1. Analyze the Asymptotic Behavior:

As $n \to \infty$, the exponential term $e^{-n}$ decays rapidly. The denominator $(n+1)^3$ grows cubically, so for large $n$, the series converges quickly because both the numerator and denominator will suppress large values of $n$.

2. Simplification or Approximation:

The series does not appear to have a closed-form solution via elementary methods. However, approximations or special functions might be used to estimate it.

If you'd like to proceed with a specific method, like numerical evaluation or further simplification through an approximation, I can calculate that as well.

Do you want me to proceed with a numerical evaluation of the sum or explore a different approach?

Here are some related questions:

1. What are the properties of rapidly converging series involving exponential terms?
2. How does asymptotic behavior help in understanding series convergence?
3. Can this sum be expressed in terms of known special functions like the Gamma function?
4. What techniques are used to estimate sums when no closed form exists?
5. How would this sum behave if we changed the powers of $n$ or the exponential?

Tip: When handling infinite sums, always check for convergence first before attempting simplification or computation!