Example 1

Tag 08I6, permanent. Cite it as [Bourbaki, Tag 08I6].

Functions of a Real Variable, chapter VI, § 1. GENERALIZED TAYLOR EXPANSIONS

If $U$ is the identity map one has $u(\lambda) = 1$ and the operator $U$ is clearly singular; since $u_n(x) = x^n$ the formula (27) of VI, p. 278, can be written, putting $\xi = \xi - \eta$

f(x + h) = \sum_{n=0}^{\infty} \frac{1}{m!} f^{(m)}(h) x^m + \int_h^{x+h} f^{(n+1)}(t) \frac{(x + h - t)^n}{n!} dt

that is, it reduces to the Taylor formula (II, p. 62).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

VI, p. 278