The binomial theorem and π

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In this activity we investigate a generalization of the binomial theorem and its connection to an approximation of $\pi$ .

The binomial theorem states $(a+b)^n = \sum _{k=0}^n \binom {n}{k} a^{n-k}b^k.$ Why would we be interested in this?

Newton says $(1+x)^r = \sum _{k=0}^\infty \left (\frac {x^k}{k!} \prod _{m = 0}^{k-1}(r-m)\right ).$ How did Newton come up with this? Hint: Calculus!

Now we’re going to use this to approximate $\pi$ .

Come up with a function of $x$ for the semicircle of radius $1/2$ centered at $(1/2,0)$ .

Use Newton’s binomial theorem to show your function above is equal to: $x^{1/2} - \frac {x^{3/2}}{2} - \frac {x^{5/2}}{8} - \frac {x^{7/2}}{16} - \frac {5x^{9/2}}{128} - \frac {7x^{11/2}}{256} - \cdots$

Use calculus to compute the area of the shaded region:

Use proportional reasoning to compute the area of the sector below:

Use the previous problem, along with the area of a certain 30-60-90 right triangle to give a different computation of the area below.

Use your work from above to give an approximation of $\pi$ .