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We are tasked with finding the greatest lower bound (infimum) and the least upper bound (supremum) of the set of numbers:

\[
\left( \frac{(n+1)^2}{2^n} \right), \text{ where } n \in \mathbb{N}
\]

### Step 1: Analyzing the Expression
We are given the expression:

\[
f(n) = \frac{(n+1)^2}{2^n}
\]

We need to find the behavior of this sequence as \(n\) increases, and identify both the infimum and supremum of the sequence.

### Step 2: Behavior as \(n \to \infty\)
- **As \(n\) becomes large**, \(2^n\) grows exponentially, while \((n+1)^2\) grows quadratically. Therefore, for large values of \(n\), the exponential growth of the denominator dominates, causing \(f(n)\) to tend to zero.
  \[
  \lim_{n \to \infty} f(n) = 0
  \]
  Thus, **the greatest lower bound (infimum)** is 0.

### Step 3: Finding the Least Upper Bound (Supremum)
- **For small values of \(n\)**, we can compute the terms of the sequence to check for the largest value:

\[
f(1) = \frac{(1+1)^2}{2^1} = \frac{4}{2} = 2
\]
\[
f(2) = \frac{(2+1)^2}{2^2} = \frac{9}{4} = 2.25
\]
\[
f(3) = \frac{(3+1)^2}{2^3} = \frac{16}{8} = 2
\]
\[
f(4) = \frac{(4+1)^2}{2^4} = \frac{25}{16} = 1.5625
\]
\[
f(5) = \frac{(5+1)^2}{2^5} = \frac{36}{32} = 1.125
\]
 
From this, we observe that the sequence starts to decrease after \(n = 2\). The highest value appears at \(n = 2\), where \(f(2) = 2.25\).

Thus, **the least upper bound (supremum)** is \(2.25\).

### Final Answer:
- The **greatest lower bound (infimum)** is \(0\).
- The **least upper bound (supremum)** is \(2.25\).

---

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:
1. How do you calculate the supremum and infimum of more complex sequences?
2. What role does the growth rate of the denominator play in determining the limits of a sequence?
3. How does the concept of convergence help in determining the infimum?
4. Can you find the infimum and supremum of a similar sequence with a different exponent on \(n\)?
5. How does this approach extend to non-integer sequences?

**Tip:** When analyzing sequences involving exponential and polynomial terms, always compare their growth rates to predict behavior as \(n \to \infty\).

Find the greatest lower and the least upper bounds of the set of numbers \((\frac{(n+1)^2}{2^n}), \text{ where } n \in \mathbb{N}\).

This problem explores the behavior of the sequence \(\frac{(n+1)^2}{2^n}\), where \(n \in \mathbb{N}\), to find its supremum and infimum. It demonstrates the interaction of exponential and polynomial growth rates, and how the sequence converges as \(n\) increases.

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