  We introduce a calculus of extensional resource terms. These are resource
terms \`a la Ehrhard-Regnier, but in infinite $\eta$-long form, while retaining
a finite syntax and dynamics: in particular, we prove strong confluence and
normalization.
  Then we define an extensional version of Taylor expansion, mapping ordinary
$\lambda$-terms to sets (or infinite linear combinations) of extensional
resource terms: just like for ordinary Taylor expansion, the dynamics of our
resource calculus allows to simulate the $\beta$-reduction of $\lambda$-terms;
the extensional nature of expansion shows in that we are also able to simulate
$\eta$-reduction.
  In a sense, extensional resource terms form a language of (non-necessarily
normal) finite approximants of Nakajima trees, much like ordinary resource
terms are approximants of B\"ohm-trees. Indeed, we show that the equivalence
induced on $\lambda$-terms by the normalization of extensional Taylor-expansion
is nothing but $H^*$, the greatest consistent sensible $\lambda$-theory.
  Taylor expansion has profoundly renewed the approximation theory of the
$\lambda$-calculus by providing a quantitative alternative to order-based
approximation techniques, such as Scott continuity and B\"ohm trees.
Extensional Taylor expansion enjoys similar advantages: e.g., to exhibit models
of $H^*$, it is now sufficient to provide a model of the extensional resource
calculus. We apply this strategy to give a new, elementary proof of a result by
Manzonetto: $H^*$ is the $\lambda$-theory induced by a well-chosen reflexive
object in the relational model of the $\lambda$-calculus.
