"In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point."
A Taylor series is a representation of a function as an infinite sum of terms.
Power Series: A power series is a representation of a function as an infinite sum of terms, where each term is a power of a variable multiplied by a coefficient.
Radius and Interval of Convergence: The radius and interval of convergence of a power series determine the values of the variable for which the series converges.
Taylor Series: A Taylor series is a special type of power series that is used to approximate a function as an infinite sum of terms involving the function's derivatives evaluated at a particular point.
Maclaurin Series: A Maclaurin series is a Taylor series that is centered at zero.
Convergence Tests: To determine the radius and interval of convergence, a series of convergence tests can be used, including the ratio and root test, the alternating series test, and the integral test.
Taylor Polynomials: A Taylor polynomial is a polynomial approximation of a function obtained by truncating its Taylor series after a finite number of terms.
Remainder Estimation: The remainder term in a Taylor series expansion measures the error made by truncating the series after a finite number of terms, and can be estimated using different methods such as Lagrange's or the integral remainder formula.
Applications of Taylor Series: Taylor series have many practical applications, including in engineering, physics, and finance, for tasks such as optimization, numerical integration, and solving differential equations.
Maclaurin Series: This is a special case of Taylor series in which the center of expansion is set to zero. It is used to approximate a function around the point x=0.
Taylor Series for Polynomials: If we take a polynomial function of degree n, its Taylor series will be exactly the same as the polynomial itself.
Taylor Series for Exponential Function: The Taylor series of e^x at a = 0 is given by 1 + x + x^2/2! + x^3/3! + ....
Taylor Series for Trigonometric Functions: The Taylor series for sine and cosine functions are given by sin(x) = x: X^3/3! + x^5/5! - ... and cos(x) = 1 - x^2/2! + x^4/4! - ... respectively.
Binomial Series: The binomial series is a special case of the Taylor series that arises when we expand a function of the form (1+x)^n. It is given by (1+x)^n = 1 + nx + n(n-1)x^2/2! + n(n-1)(n-2)x^3/3! + ....
Fourier Series: The Fourier series is a way to represent a periodic function as a sum of trigonometric functions. It has great applications in signal processing and engineering.
"Taylor series are named after Brook Taylor, who introduced them in 1715."
"A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century."
"The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function."
"Taylor polynomials are approximations of a function, which become generally more accurate as n increases."
"Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations."
"For most common functions, the function and the sum of its Taylor series are equal near this point."
"Yes, a function may differ from the sum of its Taylor series, even if its Taylor series is convergent."
"A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x."
"This implies that the function is analytic at every point of the interval (or disk)."
"Taylor series are named after Brook Taylor, who introduced them in 1715."
"Taylor series are approximations of functions, used to represent functions as infinite sums."
"Taylor polynomials are approximations of a function, which become generally more accurate as n increases."
"Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations."
"A function may differ from the sum of its Taylor series, even if its Taylor series is convergent."
"A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered."
"The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function."
"A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x."
"If a function is analytic at a point, it implies that the function is analytic at every point of the interval (or disk)."
"Colin Maclaurin made extensive use of this special case of Taylor series in the mid-18th century."