The Taylor expansion of λ-terms, as introduced by Ehrhard and Regnier, expresses a λ-term as a series of multi-linear terms, which capture bounded computations. Normal forms of Taylor expansions give a notion of infinitary normal forms, refining the notion of Böhm trees in a quantitative setting. We give the algebraic conditions over a set of normal simple terms which characterize the property of being the normal form of the Taylor expansion of a λ-term. From this full completeness result, we give further conditions which semantically describe normalizable and total λ-terms.