To determine whether the series

$\sum_{n=1}^{\infty} \frac{n^2}{n^3 - n^2 + 1}$

is convergent or not, we can apply the Limit Comparison Test or Asymptotic Analysis.

Step 1: Simplify the general term

First, let's examine the asymptotic behavior of the term $\frac{n^2}{n^3 - n^2 + 1}$ as $n \to \infty$.

For large $n$, the term $n^3$ will dominate in the denominator, so we approximate:

$n^3 - n^2 + 1 \sim n^3 \quad \text{for large } n.$

Thus, the general term behaves like:

$\frac{n^2}{n^3 - n^2 + 1} \sim \frac{n^2}{n^3} = \frac{1}{n}.$

Step 2: Use the Limit Comparison Test

The term $\frac{1}{n}$ is similar to the harmonic series, which is known to diverge. So, we use the Limit Comparison Test with the harmonic series $\sum \frac{1}{n}$.

We compute the limit of the ratio of the terms:

$\lim_{n \to \infty} \frac{\frac{n^2}{n^3 - n^2 + 1}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n^2}{n^3 - n^2 + 1} \times n = \lim_{n \to \infty} \frac{n^3}{n^3 - n^2 + 1}.$

Simplifying this expression:

$\lim_{n \to \infty} \frac{n^3}{n^3 - n^2 + 1} = \lim_{n \to \infty} \frac{1}{1 - \frac{1}{n} + \frac{1}{n^3}} = 1.$

Since the limit is a positive finite number (1), by the Limit Comparison Test, the given series behaves similarly to the harmonic series.

Step 3: Conclusion

The harmonic series $\sum \frac{1}{n}$ diverges. Therefore, by the Limit Comparison Test, the given series:

$\sum_{n=1}^{\infty} \frac{n^2}{n^3 - n^2 + 1}$

diverges as well.

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Would you like further details on any part of this solution? Here are some related questions you might find interesting:

1. How do we apply the Integral Test to determine convergence of a series?
2. What is the Comparison Test, and how is it different from the Limit Comparison Test?
3. Can a series converge if its general term does not approach zero?
4. How do we handle series with factorial terms (e.g., $\sum \frac{n!}{n^n}$)?
5. What are the common methods to prove the convergence of series involving powers of $n$?

Tip: When analyzing series, always start by comparing the given term to simpler series whose convergence behavior is already known (e.g., geometric, p-series).