Dilogarithm of One

Theorem

$\map {\Li_2} 1 = \map \zeta 2$

where:

$\map {\Li_2} x$ is the dilogarithm function of $x$

$\map \zeta 2$ is the Riemann $\zeta$ function of $2$.

Proof

\(\ds \map {\Li_2} z\)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}\)

Power Series Expansion for Spence's Function

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^\infty \frac {1^n} {n^2}\)

$z := 1$

\(\ds \)

\(=\)

\(\ds \map \zeta 2\)

Basel Problem

$\blacksquare$