Taylor polynomials and Taylor series

The derivative gives the best possible linear approximation, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function f(x) at the point x0 is a linear polynomial a + b(x - x0), and it may be possible to get a better approximation by considering a quadratic polynomial a + b(x - x0) + c(x - x0)². Still better might be a cubic polynomial a + b(x - x0) + c(x - x0)² + d(x - x0)³, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients a, b, c, and d that makes the approximation as good as possible.
For a, the best possible choice is always f(x0), and for b, the best possible choice is always f'(x0). For c, d, and higher-degree coefficients, these coefficients are determined by higher derivatives of f. c should always be f''(x0)/2, and d should always be f'''(x0)/3!. Using these coefficients gives the Taylor polynomial of f. The Taylor polynomial of degree d is the polynomial of degree d which best approximates f, and its coefficients can be found by a generalization of the above formulas. Taylor’s theorem gives a precise bound on how good the approximation is. If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f.
The limit of the Taylor polynomials is an infinite series called the Taylor series. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called analytic functions . It is impossible for functions with discontinuities or sharp corners to be analytic, but there are smooth functions which are not analytic.

Taken From http://en.wikipedia.org/wiki/Differential_calculus#Calculus_of_variations