  The signature of a $d$-dimensional Brownian motion is a sequence of iterated
Stratonovich integrals along the Brownian paths, an object taking values in the
tensor algebra over $\RR^{d}$. In this note, we derive the exact rate of
convergence for the expected signatures of piecewise linear approximations to
Brownian motion. The computation is based on the identification of the set of
words whose coefficients are of the leading order, and the convergence is
concentrated on this subset of words. Moreover, under the choice of projective
tensor norm, we give the explicit value of the leading term constant.
