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 \(\pi \).
2025-01-06 15:58:45