# Power Series Expansion for Spence's Function

## Theorem

$\ds \map {\Li_2} z = \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$

This converges for $\size z \le 1$.

## Proof

 $\ds \map {\Li_2} z$ $=$ $\ds -\int_0^z \frac {\map \ln {1 - t} } t \rd t$ Definition of Spence's Function $\ds$ $=$ $\ds -\int_0^z \frac 1 t \sum_{n \mathop = 1}^\infty \paren {-\frac {t^n} n} \rd t$ Corollary to Power Series Expansion for $\map \ln {1 + x}$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \int_0^z \frac {t^{n - 1} } n \rd t$ Fubini's Theorem $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {z^n} {n^2}$ Primitive of Power

$\blacksquare$