Riemann Zeta Function at Even Integers/Lemma/Proof 2

Lemma

Let $x \in \R$ be such that $\size x < 1$.

Then:

$\ds \pi x \cot {\pi x} = 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$

where $\zeta$ denotes the Riemann zeta function.

Proof

From Laurent Series Expansion for Cotangent Function:

$\ds \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$

where:

$z \in \C$ such that $\cmod z < 1$

$\zeta$ is the Riemann Zeta function.

Letting $x \in \R$ replace $z$, and multiplying through by $x$:

\(\ds \pi x \cot \pi x\)

\(=\)

\(\ds x \paren {\frac 1 x - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1} }\)

\(\ds \)

\(=\)

\(\ds 1 - 2 x \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n - 1}\)

\(\ds \)

\(=\)

\(\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}\)

$\blacksquare$