Taylor Series and Remainder Estimates

These flashcards cover important terms and concepts related to Taylor series, their approximations, and error estimation based on the lecture notes.

Taylor polynomial

A polynomial that approximates a function using derivatives at a single point.

Remainder estimate

An estimate of the error or difference between the actual function value and the Taylor polynomial.

Maclaurin series

A Taylor series centered at zero, which represents a function as a power series.

Pathological example

An example that is unusual or not typical, often used to illustrate a particular point in mathematics.

Convergence of Taylor series

The property that a Taylor series approaches the value of the function as more terms are added.

Taylor remainder

The difference between the actual value of the function and the value predicted by the Taylor polynomial.

Taylor's inequality

A tool used to bound the remainder of a Taylor series.

Bounded derivatives

The property indicating that the derivatives of a function are limited in magnitude within a certain interval.

Squeeze theorem

A theorem stating that if a function is squeezed between two others that both converge to the same limit, then it too converges to that limit.

Error estimation

The process of determining how close an approximation is to the actual value.

Degree of a Taylor polynomial

The highest power of (x-a) in the polynomial, which affects the accuracy of the approximation.

Interval of convergence

The range of values for which a Taylor series converges to the function it represents.

Continuous function

A function that has no breaks, jumps, or holes in its domain.

Derivatives at zero

The values of a function's derivatives evaluated at x=0, which are crucial for forming the Maclaurin series.

Exponential function

A mathematical function of the form e^x, which is used frequently in Taylor series.

Bound for the remainder

A specific value that the remainder must not exceed, ensuring the Taylor polynomial's accuracy.