Example 7.15.  Find the first few terms of the Maclaurin series for  ,  if  ,  and then compute .  

Explore Solution 7.15.

Method (i).  Use the formula    and recursively solve for the coefficients .

The coefficients are

Compare with Mathematica's Taylor expansion:

Method (ii).  Use Mathematica to expand sec(z) in a Taylor series about z = 0.

Find the fourth derivative of the series and evaluate it to compute .  

Method (iii).  Use division of power series.  For this problem we have  1 = cos(z) S(z)  where S(z) is the desired Maclaurin series.  Mathematica is used to expand cos(z) up to degree n, and solve the resulting equation for coefficients of S(z) up to n.

Remark. This is the same as the first few terms in the Maclaurin series that was computed using methods (i).

(c) 2006 John H. Mathews, Russell W. Howell