Math Problem Statement

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

Solution

To show that the series

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

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


1. Expression of the Series

The given series is:

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

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


2. Key Convergence Test

The series involves oscillatory terms (sin(nx)\sin(nx)) divided by nn, so we will use Dirichlet's Test for convergence. The test states:

If:

  1. {an}\{a_n\} is a sequence of bounded terms, e.g., anM|a_n| \leq M,
  2. bn=1nb_n = \frac{1}{n} satisfies bn0b_n \to 0 and {bn}\{b_n\} is monotonic,
  3. The partial sums Ak=n=1kanA_k = \sum_{n=1}^k a_n are bounded,

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


3. Applying Dirichlet's Test

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

  • sin(nx)\sin(nx) is clearly bounded: an=sin(nx)1|a_n| = |\sin(nx)| \leq 1, satisfying condition (1).
  • bn=1n0b_n = \frac{1}{n} \to 0 as nn \to \infty, and bnb_n is monotonic, satisfying condition (2).
  • The partial sums Ak=n=1ksin(nx)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 xx.

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


4. Conclusion

By Dirichlet's Test, the series n=1sin(nx)n\sum_{n=1}^\infty \frac{\sin(nx)}{n} converges for each fixed x(0,2π)x \in (0, 2\pi). Therefore, it is pointwise convergent on the interval (0,2π)(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 xRx \in \mathbb{R} instead of just (0,2π)(0, 2\pi)?
  4. What happens to this series' convergence if we replace sin(nx)\sin(nx) with cos(nx)\cos(nx)?
  5. How can we visualize the partial sums of this series for a fixed xx?

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

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Math Problem Analysis

Mathematical Concepts

Series convergence
Pointwise convergence
Oscillatory functions

Formulas

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

Theorems

Dirichlet's Test

Suitable Grade Level

Undergraduate Mathematics