  We study approximations for the L\'evy area of Brownian motion which are
based on the Fourier series expansion and a polynomial expansion of the
associated Brownian bridge. Comparing the asymptotic convergence rates of the
L\'evy area approximations, we see that the approximation resulting from the
polynomial expansion of the Brownian bridge is more accurate than the
Kloeden-Platen-Wright approximation, whilst still only using independent normal
random vectors. We then link the asymptotic convergence rates of these
approximations to the limiting fluctuations for the corresponding series
expansions of the Brownian bridge. Moreover, and of interest in its own right,
the analysis we use to identify the fluctuation processes for the
Karhunen-Lo\`eve and Fourier series expansions of the Brownian bridge is
extended to give a stand-alone derivation of the values of the Riemann zeta
function at even positive integers.
