# Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be $$g\left(x\right)=2\sum^{\infty}_{n=1}\dfrac{\left(-1\right)^{n+1}}{n}\sin\left(nx\right)$$

I now need to determine and explain whether the Fourier series converges at $x=\pi$ but I'm not sure how.

If it involves finding the left/right limits/derivatives, I'm not sure how to find these for a Fourier Series.

• What is $\sin(n\pi)$? – David C. Ullrich May 24 '16 at 18:57

for $x=\pi$, the series is $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(n\pi) = \sum_{n=1}^{\infty} 0 = 0$$
so the series converges to 0 at $\pi$. We expect this because 0 is the midpoint between $g(-\pi)$ and $g(\pi)$