It has recently been proved
that a continuous path of bounded variation in $\R^d$ can be
characterised in terms of its transform into a sequence of iterated
integrals called the signature of the path. The signature takes its
values in an algebra and always has a logarithm. In this paper we
study the radius of convergence of the series corresponding to this
logarithmic signature for the path. This convergence can be
interpreted in control theory (in particular, the series can be used
for effective computation of time invariant vector fields whose
exponentiation yields the same diffeomorphism as a time inhomogeneous
flow) and can provide efficient numerical approximations to solutions
of SDEs. We give a simple lower bound for the radius of convergence of
this series in terms of the length of the path. However, the main
result of the paper is that the radius of convergence of the full log
signature is finite for two wide classes of paths (and we conjecture
that this holds for all paths different from straight lines).