  In this note, given a polarized algebraic manifold $(X,L)$, we define the
Donaldson-Futaki invariant for a sequence of test configurations for $(X,L)$
with exponents tending to infinity. This then allows us to define a strong
version of K-stability or K-semistability for $(X,L)$. In particular, $(X,L)$
will be shown to be K-semistable in this strong sense if the polarization class
$c_1(L)$ admits a constant scalar curvature Kaehler metric.
