In Stein's Fourier Analysis i'm having trouble Justifying the following calculations for the Fourier Coefficients in $(1.)$
$(1.)$
Let $f(\theta) = \theta$ for $ -\pi \leq \theta \leq \pi$ in the case when $n \neq 0$:
$$f(n)=\frac{1}{2 \pi}\int_{-\pi}^{\pi}\theta e^{-in \theta} d \theta = \frac{1}{2 \pi}[-\frac{\theta}{in}e^{-in \theta}]_{-\pi}^{\pi} + \frac{1}{2 \pi in} \int_{-\pi}^{\pi}e^{-in\theta}d \theta = \frac{(-1)^{n+1}}{in} $$
Then the Fourier Series of $f$ is given by:
$$ f(n) = \sum_{n \neq 0} \frac{(-1)^{n+1}}{in}$$
$$\frac{1}{2 \pi}[-\frac{\theta}{in}e^{-in \theta}]_{-\pi}^{\pi} + \frac{1}{2 \pi in} \int_{-\pi}^{\pi}e^{-in\theta}d \theta$$ $$=-\frac{1}{2 \pi}[\frac{\pi}{in}e^{-in \pi}-\frac{-\pi}{in}e^{+in \pi}] + \frac{1}{2 \pi in}\cdot \left[ \frac{e^{-in\theta}}{{-in}}\right] ^{\pi}_{-\pi} $$ $$=-\frac{1}{2 \pi}[\frac{\pi}{in}e^{-in \pi}-\frac{-\pi}{in}e^{+in \pi}] + \frac{1}{2 \pi in}\cdot \left[ \frac{e^{-in\pi}}{{-in}}-\frac{e^{+in\pi}}{{-in}}\right] $$ Since $$ e^{n\pi i}= e^{-n\pi i}=(-1)^n$$ The second term cancels and the first term becomes: $$-\frac{(-1)^n}{in}= \frac{(-1)^{n+1}}{in}$$