Maclaurin and Taylor Polynomials

    When a Taylor series is truncated to a finite number of terms the result is a Taylor polynomial.  A Taylor series expanded about , is called a Maclarin series.  These Taylor (and Maclaurin) polynomials are used to numerically approximate functions.  We attribute much of the founding theory to Brook Taylor (1685-1731), Colin Maclaurin (1698-1746) and Joseph-Louis Lagrange (1736-1813).

Theorem (Taylor Polynomial Approximation ).  Assume that  ,  then

    ,
    
where is a polynomial that can be used to approximate  , and we write  

    .

The remainder term has the form

    ,

for some value that lies between .  The formula is referred to as the Lagrange form of the remainder.