Example 5.  Consider the function  .  
5 (a).  Find the terms up to    in the Maclaurin series for  f[x].
Solution 5 (a).

5 (a).  Find the terms up to    in the Maclaurin series for  f[x].

Remark.  If you just find the "Series" it will include a "Big "  term, which cannot be used in either evaluations or graphing, we eliminate it with the command "Normal."  The "Big   term lets us know the power of x in the "remainder.

Aside.  Of course the "full" Maclaurin series has an infinite number of terms.  Mathematica is capable of finding sums of infinite series.  

The following is just for fun !

There are two ways to get infinite sums, the old way and the new way.

The "symbolic" command is:

Now graph f[x] and the Maclaurin polynomial s[x] over the interval [-3, 3].  

The above approximation is not too good near the endpoints, this is why we want to graph f[x] and the Maclaurin polynomial s[x] over the interval [-2, 2].  

Background for part 5 (b).  Use the fact that the series is "alternating" to investigate the error for the Maclaurin polynomial of degree n = 20 over the interval  [-2.0, 2.0].  

Now look closely at the "error" when the series is used to approximate the function.  How close are were the two curves in part (a) ?

This series is "very nice" because it is alternating, and for that reason the error bound is the magnitude of the "next non-zero term" in the series.

The error bound for the entire interval  [-2.0, 2.0] is

This estimate is "conservative" and is a little larger than the "actual" maximum error which occurred at     

(c) John H. Mathews 2004