Exercise 4 Solution

(You may want to maximize your window to see the solution more clearly.)

As noted in the hint, find the expansion of  using long division:

Continuing the division, we can see that the Taylor series for  is

Integrating this term by term gives

Thus, the 10th-order Taylor polynomial (expanded about a = 0) is

Evaluating in the above expression for π gives

which is a good approximation of π to seven decimal places. Here's a better approximation:

(This approximation of π may be calculated from the above formula, but using many more terms of the necessary Taylor expansion.)