Exercise 1, § 2

Establish the formulae

here the series on the right-hand sides converge absolutely, the first for $|z| < \frac{\pi}{2}$ and the second for $|z| < \pi$ (express $\tan z$ and $1/\sin 2z$ as linear combinations of $\cot z$ and $\cot 2z$). Deduce that the numbers $\frac{2^{2n}(2^{2n} - 1)}{2n} b_{2n}$ are integers. (Use the following lemma: in two absolutely convergent series $\sum_{n=0}^{\infty} \alpha_n \frac{z^n}{n!}$, $\sum_{n=0}^{\infty} \beta_n \frac{z^n}{n!}$ the coefficients $\alpha_n$ and $\beta_n$ are integers, then, in their product written in the form $\sum_{n=0}^{\infty} \gamma_n \frac{z^n}{n!}$, the $\gamma_n$ are integers.)