Let $a=N+1 / 2$ for a positive integer $N$. Let $C_{N}$ be the anticlockwise contour defined by the square with its four vertices at $a \pm i a$ and $-a \pm i a$. Let

$$I_{N}=\oint_{C_{N}} \frac{d z}{z^{2} \sin (\pi z)}$$

Show that $1 / \sin (\pi z)$ is uniformly bounded on the contours $C_{N}$ as $N \rightarrow \infty$, and hence that $I_{N} \rightarrow 0$ as $N \rightarrow \infty$.

Using this result, establish that

$$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{2}}=\frac{\pi^{2}}{12}$$