Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
Need help with checking: $\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}\sin(nx)}{n}$
for point-wise convergence and uniform convergence of: ${-\pi} \leq x \leq {\pi}$.
You can use Abel's uniform convergence test. See here. Or you can use the following theorem:
Theorem: The Fourier series of a 2π-periodic continuous and piecewise smooth function converges uniformly.