Taylor's and Maclaurin's Series A Taylor series is an infinite series that represents a function in a specific interval using the values of the function at...
Taylor's and Maclaurin's Series
A Taylor series is an infinite series that represents a function in a specific interval using the values of the function at specific points. These points are called the Taylor coefficients. The Maclaurin series is another type of infinite series that is centered at a specific point.
Taylor series
Taylor series can be expressed in the general form as:
where:
(f(x)) is the function whose Taylor series is being expanded.
(a) is the center of the Taylor series.
(f^{(n)}(a)) is the -th derivative of (f(x)) evaluated at (a).
(n!) is the factorial of (n).
Maclaurin series
Maclaurin series is a special case of Taylor series where the center is located at (x = a). The Maclaurin series has the following form:
where (f^{(n)}(a)) is the -th derivative of (f(x)) evaluated at (a).
Examples
Consider the function (f(x) = \sin(x)). The Taylor series of (f(x)) centered at (a = 0) is:
The Maclaurin series of (f(x)) centered at (a = 0) is:
These series represent the function (f(x)) in their respective intervals