Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ and a parity argument, show that $b_{2p+1} = 0$ for all integer $p \geqslant 1$.
Using the expansion $\frac{z}{\mathrm{e}^z - 1} = \sum_{n=0}^{+\infty} \frac{b_n}{n!} z^n$ and a parity argument, show that $b_{2p+1} = 0$ for all integer $p \geqslant 1$.