In this note, we explore applications of a known lemma which relates the expected exit time of Brownian motion from a simply connected domain with the power series of a conformal map into that domain. We use the lemma to calculate the expected exit time from a number of domains, and in the process describe a probabilistic method for summing certain series. In particular, we give a proof of Euler's classical result that $\zeta(2) = \pi^2/6$. We also show how the relationship between the power series and the Brownian exit time gives several immediate consequences when teamed with a deep result of de Branges concerning the coefficients of power series of normalized conformal maps. We conclude by stating an extension of the lemma in question to domains which are not simply connected.
"On the expected exit time of planar Brownian motion from simply connected domains." Electron. Commun. Probab. 16 652 - 663, 2011. https://doi.org/10.1214/ECP.v16-1653