Theorem: Taylor's Theorem (Lagrange Form of the Remainder)
Gives the Lagrange form of the Taylor remainder, providing an explicit bound on the error of a Taylor polynomial approximation.
Definition: Maclaurin Series
Defines the Maclaurin series as the special case of a Taylor series centered at zero, and lists key standard Maclaurin series.
Proposition: Using Taylor Series to Evaluate Limits
Illustrates using the Maclaurin series of a function to evaluate indeterminate limits by series substitution.
Proposition: Using Taylor Series to Approximate Integrals
Illustrates approximating non-elementary definite integrals by substituting a Maclaurin series for the integrand and integrating term by term.
Definition: Taylor Series
Defines the Taylor series of an infinitely differentiable function about a point, using its higher-order derivatives.
Definition: Taylor Polynomial
Defines the n-th degree Taylor polynomial of a function centered at c, and the remainder term measuring its approximation error.
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**False.** For $f(x) = \sin x$, the derivatives cycle as $\sin x, \cos x, -\sin x, -\cos x, \dots$, giving values at $x …
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True or False: The Taylor series of f(x)=sinxf(x) = \sin xf(x)=sinx about a=0a = 0a=0 is ∑n=0∞x2n+1(2n+1)!\displaystyle \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}n=0∑∞(2n+1)!x2n+1 (with no alternating sign).
True
False