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To show that the series 

\[
\sum_{n=1}^\infty \frac{\sin(nx)}{n}
\]

is **pointwise convergent (PC)** on \((0, 2\pi)\), let us proceed step by step:

---

### 1. Expression of the Series
The given series is:

\[
S(x) = \sum_{n=1}^\infty \frac{\sin(nx)}{n}.
\]

We aim to prove that this series converges pointwise for \(x \in (0, 2\pi)\).

---

### 2. Key Convergence Test
The series involves oscillatory terms (\(\sin(nx)\)) divided by \(n\), so we will use **Dirichlet's Test** for convergence. The test states:

If:
1. \( \{a_n\} \) is a sequence of bounded terms, e.g., \(|a_n| \leq M\),
2. \( b_n = \frac{1}{n} \) satisfies \( b_n \to 0 \) and \(\{b_n\}\) is monotonic,
3. The partial sums \( A_k = \sum_{n=1}^k a_n \) are bounded,

then the series \( \sum_{n=1}^\infty a_n b_n \) converges.

---

### 3. Applying Dirichlet's Test

Here, let \( a_n = \sin(nx) \) and \( b_n = \frac{1}{n} \).

- \(\sin(nx)\) is clearly bounded: \( |a_n| = |\sin(nx)| \leq 1 \), satisfying condition (1).
- \( b_n = \frac{1}{n} \to 0 \) as \( n \to \infty \), and \( b_n \) is monotonic, satisfying condition (2).
- The partial sums \( A_k = \sum_{n=1}^k \sin(nx) \) remain bounded. This can be shown using trigonometric identities to express sums of sine terms as bounded oscillatory functions of \(x\).

Thus, all conditions of Dirichlet's Test are satisfied.

---

### 4. Conclusion
By Dirichlet's Test, the series \( \sum_{n=1}^\infty \frac{\sin(nx)}{n} \) converges for each fixed \(x \in (0, 2\pi)\). Therefore, it is pointwise convergent on the interval \((0, 2\pi)\).

---

Let me know if you want a detailed breakdown of how to explicitly calculate or bound the partial sums, or if you have further questions!

Here are 5 related questions for deeper understanding:

1. What is the difference between pointwise and uniform convergence for a series of functions?
2. How does Dirichlet's Test differ from Abel's Test for convergence?
3. Can we extend this analysis to \(x \in \mathbb{R}\) instead of just \((0, 2\pi)\)?
4. What happens to this series' convergence if we replace \(\sin(nx)\) with \(\cos(nx)\)?
5. How can we visualize the partial sums of this series for a fixed \(x\)?

**Tip:** When testing convergence for oscillatory series, always consider boundedness of partial sums and monotonic decay of coefficients as primary criteria.

Să se arate că seria \( \sum_{n=1}^\infty \frac{\sin(nx)}{n} \) este PC pe \((0, 2\pi)\).

Learn how to prove the pointwise convergence of the series \( \sum_{n=1}^\infty \frac{\sin(nx)}{n} \) on the interval \((0, 2\pi)\). This detailed solution uses Dirichlet's Test and explains key concepts like oscillatory functions and bounded partial sums. Perfect for undergraduate mathematics.

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