Module for Maclaurin and Taylor Series

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Background. 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   and is a fixed value.  

If  ,  then

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

    .

The error term has the form

    ,

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

Animations (Taylor and Maclaurin Polynomial Approximation  Taylor and Maclaurin Polynomial Approximation).  
Internet hyperlinks to animations.

Example 1.  Consider the function  .  
1 (a).  Find the terms up to    in the Maclaurin series for  f[x].
1 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 10 over the interval  [-0.5, 0.5].  
1 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].

Solution 1 (a).

Solution 1 (b).

Solution 1 (c).

Mathematical notation.  Mathematica has adopted the notation    for the natural logarithm.  This can be illustrated by using either differentiation or integration.  Since    starts with the upper case letter  L,  the word    is a "reserved word."

Example 2.  Consider the function  .  
2 (a).  Find the terms up to    in the Maclaurin series for  f[x].
2 (b).  Investigate the error in the approximation over the interval [-0.8, 0.8].

Solution 2.

Example 3.  Consider the function  .  
3 (a).  Find the terms up to    in the Maclaurin series for  f[x].
3 (b).  Investigate the error term for the Maclaurin polynomial of degree n = 20 over the interval  [-2.0, 2.0].  
3 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].

Solution 3 (a).

Solution 3 (b).

Solution 3 (c).

Old Lab Project (Maclaurin Polynomials  Maclaurin Polynomials).  Internet hyperlinks to an old lab project.  

Old Lab Project (Taylor Polynomials  Taylor Polynomials).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

Maclaurin and Taylor Series  Maclaurin and Taylor Series  Internet hyperlinks to web sites and a bibliography of articles.  

Downloads (Maclaurin and Taylor Series Maclaurin and Taylor Series).  

Download this Mathematica notebook.  

(c) John H. Mathews 2003