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To discuss the convergence of the given series:

\[
\sum_{n=1}^{\infty} \frac{n^2}{\left(1 + \frac{1}{n}\right)^n}
\]

We need to analyze its behavior, particularly whether it converges or diverges.

### Step 1: Simplify the General Term
First, simplify the general term of the series:

\[
\frac{n^2}{\left(1 + \frac{1}{n}\right)^n}
\]

Recall the limit definition of the exponential function \(e\):

\[
\left(1 + \frac{1}{n}\right)^n \approx e \quad \text{as} \quad n \to \infty
\]

Thus, for large \(n\), the expression can be approximated by:

\[
\frac{n^2}{e} 
\]

### Step 2: Analyze the General Term
Now, consider the simplified form of the general term:

\[
a_n = \frac{n^2}{e}
\]

### Step 3: Apply a Convergence Test
To determine convergence, you can use the **Divergence Test** (also known as the Test for Divergence):

\[
\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n^2}{e} = \infty
\]

Since the general term \(a_n\) does not approach zero as \(n\) increases but instead grows without bound, the series diverges.

### Conclusion:
The series

\[
\sum_{n=1}^{\infty} \frac{n^2}{\left(1 + \frac{1}{n}\right)^n}
\]

**diverges**.

Would you like more details on any specific part of this solution, or do you have any questions? 

Here are some related questions you might find interesting:

1. What is the divergence test, and how does it determine the convergence of a series?
2. How does the approximation of \(\left(1 + \frac{1}{n}\right)^n\) by \(e\) play a role in the analysis of series?
3. What other tests can be used to determine the convergence of series?
4. What is the difference between conditional and absolute convergence?
5. Can the series be analyzed using the ratio or root tests for convergence?

**Tip:** Always check the limit of the general term of a series first; if it doesn't approach zero, the series diverges immediately.

Explore the convergence analysis of the series sum_n=1^∞ n^2 / ((1 + 1/n)^n). Understand the application of the Divergence Test and the limit approximation of (1 + 1/n)^n to e. Suitable for advanced undergraduate students and those interested in mathematical series analysis.

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