convergence of sum of sin(aknx)/n

The prove

The convergence of the sum of \(\sum_{n=1}^{\infty} \frac{sin(anx)}{x}\)

Conclusion

Thus, we conclude that

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

• Pointwise convergent (only) on open interval \(\left(0, 2\pi\right)\);
• Uniformly convergent on any close sub interval of \(\left(0, 2\pi\right)\).
• The series on the other hand converges for every \(x\).

More information about the theroem

More information can be found:

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