Suppose that f(x) is a function, and that all the derivatives f ', f '', f ''', etc. exist at x = a. Then the Taylor Series of f(x) is the power series
or, in sigma notation,
A Maclaurin series is a Taylor series in the case where a = 0.
The partial sums of the Taylor series are called Taylor polynomials. These can be used to approximate the function in the neighborhood of x = a.
Find the Taylor polynomials for the function
f(x) = sin(x)
around x = 0.