The problem requires finding the supremum and infimum of the set $S$ defined as follows:

$S = \left\{ -\frac{1}{n} + \left(1 + (-1)^n \right) n^2 \mid n \in \mathbb{N} \right\}$

To analyze the set $S$, let's break down the expression given for each $n$:

1. Expression Analysis: The expression for each element in $S$ is: $-\frac{1}{n} + \left(1 + (-1)^n \right) n^2$ where $(-1)^n$ alternates between $1$ (for even $n$) and $-1$ (for odd $n$).

2. Cases Based on Parity of $n$:

   • When $n$ is odd:
     • $(-1)^n = -1$
     • The expression simplifies to: $-\frac{1}{n} + \left(1 + (-1) \right) n^2 = -\frac{1}{n}$
   • When $n$ is even:
     • $(-1)^n = 1$
     • The expression simplifies to: $-\frac{1}{n} + \left(1 + 1 \right) n^2 = -\frac{1}{n} + 2n^2$

3. Finding Supremum and Infimum:

   • For odd $n$, the values approach $0$ as $n$ increases, but remain negative. Hence, the infimum of these values will be slightly below zero, depending on the smallest odd $n$.

   • For even $n$, the values become very large as $n$ increases due to the $2n^2$ term. Thus, the supremum of the set $S$ will be the largest value produced by even $n$, which can go to infinity as $n \to \infty$.

   • Supremum: There is no upper bound, so the supremum is $+\infty$.

   • Infimum: The infimum will be the limit approached by the values for odd $n$, which is $0$ from below. Therefore, the infimum of $S$ is $0$.

In summary:

• Supremum of $S$: $+\infty$
• Infimum of $S$: $0$

Would you like more details or have any questions?

Here are some further related questions:

1. How do limits help in determining the infimum and supremum in sequences?
2. What would happen if we modified $(-1)^n$ to something else, like a constant?
3. How can we formally prove that a sequence diverges to infinity?
4. What are some real-world applications of finding infimum and supremum?
5. How would the solution change if the domain was restricted to only even or only odd $n$?

Tip: When finding supremum or infimum, look at limiting behavior and consider cases based on parameters (like $n$ being odd or even).