= Cotangent coefficient proof of the even zeta values
The <cotangent partial-fraction expansion> gives, on $0<|z|<1$,
$$
\pi\cot(\pi z)=z^{-1}-2\sum_{r\geq1}\zeta(2r)z^{2r-1}.
$$
The <Bernoulli number> <generating function> gives the same <Laurent series> in another form:
$$
\pi\cot(\pi z)=\pi i+\frac{2\pi i}{e^{2\pi iz}-1}
=z^{-1}+\sum_{r\geq1}\frac{B_{2r}(2\pi i)^{2r}}{(2r)!}z^{2r-1}.
$$
The $B_1$ term cancels the separate $\pi i$; the higher odd <Bernoulli numbers> vanish. Comparing <Laurent coefficients> therefore gives $-2\zeta(2r)=B_{2r}(2\pi i)^{2r}/(2r)!$, which is the stated real formula because $i^{2r}=(-1)^r$. The <cotangent partial-fraction expansion> justifies the convergent expansion before the <coefficient> comparison; a finite plot of <partial sums> alone would not prove the infinite identity.
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