  In this article, an exact series expansion for the Dottie number (solution of
the equation $\cos(x) = x$) is presented. Its derivation consists in combining
the Kaplan representation of the Dottie number as a series in odd powers of
$\pi$, with a series for $\pi^{2k+1}$ ($k\geq 1$) obtained using the Fa\`a di
Bruno formula.
