  The {\it forward-backward algorithm} is a powerful tool for solving
optimization problems with a {\it additively separable} and {\it smooth} + {\it
nonsmooth} structure. In the convex setting, a simple but ingenious
acceleration scheme developed by Nesterov has been proved useful to improve the
theoretical rate of convergence for the function values from the standard
$\mathcal O(k^{-1})$ down to $\mathcal O(k^{-2})$. In this short paper, we
prove that the rate of convergence of a slight variant of Nesterov's
accelerated forward-backward method, which produces {\it convergent} sequences,
is actually $o(k^{-2})$, rather than $\mathcal O(k^{-2})$. Our arguments rely
on the connection between this algorithm and a second-order differential
inclusion with vanishing damping.
