Use two formulas to approximate $\pi$

Question

Hi guys. Any hints on part c? How are we going to approximate $\pi$ by computing $P_k$ and $p_k$

Answer 1

The whole idea is that $p_k \to \pi$ as $k \to \infty$: it is known that $\frac{\sin(x)}{x} \to 1$ as $x \to 0$, so

$$ \frac{\sin\frac{\pi}{k}}{\frac{\pi}{k}} \to 1 $$

therefore $k \sin \frac{\pi}{k} \to \pi$.

Edit, to expand on my answer a bit:

Similarly, one can show that $P_k$ also converges to $\pi$. Point (b) shows that $P_{2k}$ is a harmonic average of $p_k$ and $P_k$, while $p_{2k}$ is their geometric average. Therefore, we have $p_k < p_{2k} < P_{2k} < P_k$. Since both $p_k$ and $P_k$ converge to $\pi$, it follows from these inequalities that $p_{2^k} < \pi < P_{2^k}$ for all $k$. Therefore, if $P_{2^k} - p_{2^k} < 10^{-4}$, both $P_{2^k}$ and $p_{2^k}$ are within $10^{-4}$ from $\pi$.

Answer 2

For part (c), just calculate $P_k$ and $p_k$ for $k$ equal to a power of 2. You already have $P_4$ and $p_4$, so now apply (b) with $k=4$ to get $P_8$ and $p_8$, then apply (b) again with $k=8$ to get $P_{16}$ and $p_{16}$, and so on. Stop when the difference between these is small enough.