Example 4.  Consider the function  .  
4 (c).  Find the terms up to    in the Maclaurin series and see how close it approximates  f[x].
Solution 4 (c).

4 (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 .

Question.  Do we have a "good approximation" on the interval   ?

Aside.  Mathematica actually computes the higher derivatives of f[x] up to the order 50 when calculating the Taylor series. Suppose you had to find the formula for the first 20 derivatives of f[x].  Could "you" do it ?  Would you want to do it ?

Warning.  Be sure that you leave the command "Together" in the print statement.  Otherwise the size of the output for the derivatives is 79MB, which might crash your computer.

(c) John H. Mathews 2004