Example 1.  Consider the function  .  
1 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 1 (c).

1 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].

Go ahead "enjoy" and add terms in the series up to , then plot the functions over the interval [-0.8, 0.8].

Aside.  Mathematica actually computes the higher derivatives of f[x] when calculating the Taylor series. Suppose you had to find the formula for the first 10 derivatives of f[x].  Could "you" do it ?  Would you want to do it ?

Aside.  The function    is "infinitely differentiable", that is it has a derivatives of all orders.  And the graph looks really smooth and nice for "all x."

So, why is the Maclaurin series severely restricted to the interval (-1,1).  The answer lies in the study of complex numbers, we must  look at the denominator and see where it vanishes.

The values   are called singularities of  f[x]  and the radius of convergence is the distance from the origin to the closest one.  But this topic must be left to another course called "complex analysis" which is offered each spring.  All computer algebra systems such as Mathematica or Maple or Derive do there underlying computations using complex numbers.  It is the mathematicians way to do it, and it is really fun to work with complex functions.

(c) John H. Mathews 2004