Chapter 6.4, Problem 85E

Challenge Yourself

Continue the calculation of $\pi$ in Example 2 (a) for 20 more terms. How does this approximation compare with the one we obtained in Example 2?

Example 2 Approximating $\mathit{\pi },\mathit{ }\mathit{\varnothing },\mathit{ }\mathbf{a}\mathbf{n}\mathbf{d}\mathit{ }\mathit{e}$

Although $\pi$ = 3.141592654…, $\varnothing$ = 1.618033989 …, and e = 2.7182818284 … are irrational numbers and therefore have nonrepeating decimal expansions, there are methods for computing good approximations to these numbers.

a. Approximate $\pi$ by using the first 20 terms of the following series:

$4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\frac{4}{13}$