Definition:Taylor Series/Remainder/Cauchy Form

Definition

Let $f$ be a real function which is smooth on the open interval $\openint a b$.

Let $\xi \in \openint a b$.

Consider the remainder of the Taylor series at $x$:

$\ds \map {R_n} x = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$

The Cauchy form of the remainder $R_n$ is given by:

$R_n = \dfrac {\paren {x - \eta}^n} {n!} \paren {x - \xi} \map {f^{\paren {n + 1} } } \eta$

where $\eta \in \closedint \xi x$.

Also see

• Definition:Lagrange Form of Remainder of Taylor Series

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Taylor Series for Functions of One Variable: $20.3$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.26$: Cauchy ($\text {1789}$ – $\text {1857}$)
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Taylor Series for Functions of One Variable: $22.3.$

• Weisstein, Eric W. "Cauchy Remainder." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CauchyRemainder.html