Taylor's theorem with integral remainder
, giving and, for uniformly bounded , convergence to the Taylor series.
Theorem 4.27 (Taylor's Theorem: Integral Remainder)
Suppose . Let be as before. Then
Proof
Using integration by parts repeatedly,
which is exactly the Taylor remainder.
Consequences
By the Extreme Value Theorem, , so
hence as for fixed . If moreover , then as for all , which would mean is analytic at .
The theorem generalises to where contains the line segment . In , differentiability implies smoothness and then analyticity, with estimates on that give convergence of to as .
The integral form also recovers the other remainder forms via the Cauchy Mean Value Theorem for integrals:
- Lagrange form — taking , there is with
- Cauchy form — taking , there is with
Related
Stated in
- Theorem 4.27 (Taylor's Theorem: Integral Remainder)§4.4 Integration and Differentiation