Exercise 8

In this exercise we consider the definite integral $$\begin{gathered} \int_0^1 \frac{4}{1+x^2} \, dx \end{gathered}$$ from two different perspectives.

1. First, find the Taylor series for $\frac{4}{1+x^2}$ and use it to evaluate $$\begin{gathered} \int_0^1 \frac{4}{1+x^2} \, dx \end{gathered}$$ as an infinite series of real numbers.

2. Observe that your result in (a) is an alternating series. Estimate the value of that alternating series to within $0.01$. How many terms of the series are needed to do so?

3. Recall that $\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}$. Use this fact and the First Fundamental Theorem of Calculus to evaluate $$\begin{gathered} \int_0^1 \frac{4}{1+x^2} \, dx \end{gathered}$$ exactly.

4. How are your results in (a) and (c) connected? What famous number can we now estimate using an alternating series? How many terms of the series were needed to ensure the first two digits of the famous number are accurate?