We are tasked to determine whether the series

$\sum_{n=1}^\infty \sin\left(\frac{1}{n^2}\right)$

converges or diverges and to justify the conclusion.

Analysis:

1. Behavior of the Term:

   • For large $n$, $\frac{1}{n^2}$ approaches 0.
   • Using the Taylor expansion for $\sin(x)$ around 0: \sin(x) \approx x \quad \text{(for small x)}.
   • Hence, for large $n$, $\sin\left(\frac{1}{n^2}\right) \approx \frac{1}{n^2}.$

2. Comparison to a Known Series:

   • The series $\sum_{n=1}^\infty \frac{1}{n^2}$ is a p-series with $p = 2 > 1$, and it converges.
   • Since $\sin\left(\frac{1}{n^2}\right) \sim \frac{1}{n^2}$, we suspect $\sum_{n=1}^\infty \sin\left(\frac{1}{n^2}\right)$ converges.

3. Limit Comparison Test:

   • Let $a_n = \sin\left(\frac{1}{n^2}\right)$ and $b_n = \frac{1}{n^2}$.
   • Compute: $\lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\sin\left(\frac{1}{n^2}\right)}{\frac{1}{n^2}}.$
   • For small $x$, $\sin(x) \approx x$, so: $\frac{\sin\left(\frac{1}{n^2}\right)}{\frac{1}{n^2}} \to 1 \quad \text{as } n \to \infty.$
   • Since the limit is a positive finite constant ($1$), and $\sum b_n = \sum \frac{1}{n^2}$ converges, the Limit Comparison Test implies $\sum a_n$ also converges.

Conclusion:

The series $\sum_{n=1}^\infty \sin\left(\frac{1}{n^2}\right)$ converges. This conclusion is justified by approximating the terms and applying the Limit Comparison Test.

Do you want a detailed explanation of the Limit Comparison Test or the derivation of the Taylor expansion? Here are related questions to expand on this:

1. How does the Taylor series approximation for $\sin(x)$ work, and what are its higher-order terms?
2. Why does the series $\sum \frac{1}{n^2}$ converge, and how does this relate to p-series?
3. Can the Direct Comparison Test be applied here instead of the Limit Comparison Test?
4. How does the behavior of $\sin(x)$ influence series convergence for different powers of $n$?
5. What happens if we replace $\sin\left(\frac{1}{n^2}\right)$ with $\sin\left(\frac{1}{n}\right)$?

Tip: When approximating terms in series, always check the validity of using asymptotic approximations and whether a formal convergence test is necessary.