Math Problem Statement
Solution
To show that the series
is pointwise convergent (PC) on , let us proceed step by step:
1. Expression of the Series
The given series is:
We aim to prove that this series converges pointwise for .
2. Key Convergence Test
The series involves oscillatory terms () divided by , so we will use Dirichlet's Test for convergence. The test states:
If:
- is a sequence of bounded terms, e.g., ,
- satisfies and is monotonic,
- The partial sums are bounded,
then the series converges.
3. Applying Dirichlet's Test
Here, let and .
- is clearly bounded: , satisfying condition (1).
- as , and is monotonic, satisfying condition (2).
- The partial sums remain bounded. This can be shown using trigonometric identities to express sums of sine terms as bounded oscillatory functions of .
Thus, all conditions of Dirichlet's Test are satisfied.
4. Conclusion
By Dirichlet's Test, the series converges for each fixed . Therefore, it is pointwise convergent on the interval .
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:
- What is the difference between pointwise and uniform convergence for a series of functions?
- How does Dirichlet's Test differ from Abel's Test for convergence?
- Can we extend this analysis to instead of just ?
- What happens to this series' convergence if we replace with ?
- How can we visualize the partial sums of this series for a fixed ?
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
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