Example 1.  Consider the function  .  
1 (a).  Find the terms up to    in the Maclaurin series for  f[x].
Solution 1 (a).

1 (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.  First sum the series  .  The "telegraph" command is:

The "symbolic" command is:

Summation is located on the BasicInput palette.
Now sum the power series  .  

Now graph f[x] and the Maclaurin polynomial s[x] over the interval [-1, 1].  

Notice that there is a significant amount of error near  .  Let's investigate a smaller interval.
Graph  f[x]  and the Maclaurin polynomial  s[x]  over the interval  [-0.5, 0.5].  

Background for part 1 (b).  Use the fact that the series is "alternating" to investigate the error for the Maclaurin polynomial of degree n = 10 over the interval  [-0.5, 0.5].  

Now look closely at the "error" when the series is used to approximate the function.  It becomes infinite near x = 1.  For this reason we work on smaller intervals, in this case we chose  [-0.5, 0.5].  How close are were the two curves in part (a) ?

Observation.  We used the "Simplify" command to form e[x], without this command it would have looked different.

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

The error bound for the entire interval  [-0.5, 0.5] is

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

(c) John H. Mathews 2004