For a smooth projective variety X and a coherent sheaf S on it, you can consider the projective cone $\pi:\mathbb{P}S \rightarrow X$. You can consider $\mathcal{O}(1)$ on $\mathbb{P}S$ and define the Segre series a la Fulton:
$$
s(S,t) = \pi_*\left(\frac {1}{t-c_1(\mathcal{O}(1))}\right) \in A^*(X)
$$
I can also take a finite projective resolution of $S$ by vector bundles, and ask how to express $s(S,t)$ in terms of the Chern classes of these bundles.
Best Answer
For a vector bundle $E$ its Segre class is defined as $$ s(E) = \prod(1+x_i)^{-1}, $$ where $x_i$ are Chern roots of $E$. Because of this it is clear that Segre class is multiplicative in short exact sequences. Therefore for a coherent sheaf $S$ if $$ 0 \to E_n \to \dots \to E_2 \to E_1 \to E_0 \to S \to 0 $$ is a locally free resolution then $$ s(S) = s(E_0)s(E_1)^{-1}s(E_2)\cdots(E_n)^{(-1)^n}. $$