I am not exactly sure how to approach the following question. The question states that we are given that $A(n)$ represents the perimeter of a regular n-gon that has a circle of radius $1$ inside of it, and $B(n)$ represents the perimeter of a regular n-gon that is inside a circle with radius $1$.
The question, then asks us to:
A) Compute $A(4)$ and $B(4)$
B) State what we think $\lim_{n \to \infty} A(n)$ and $\lim_{n \to \infty} B(n)$ will be.
C) What estimates do $A(4)$ and $B(4)$ give us for a famous mathematical constant ?
For A) I have got that $A(4)$ = 8, and $B(4) = 4\sqrt{2}$, which I'm not sure to be correct. For parts B) and C), I don't really know how to tackle these questions.
Any help would be greatly appreciated!
Best Answer
Your values for $A(4)$ and for $B(4)$ are correct. And both $\lim_{n\to\infty}A(n)$ and $\lim_{n\to\infty}B(n)$ are equal to the perimer of a circle with radius $1$; that is, they are equal to $2\pi$. So, since $\bigl(A(n)\bigr)_ {n\in\mathbb N}$ is decreasing and $\bigl(B(n)\bigr)_ {n\in\mathbb N}$ is increasing, $4\sqrt2<2\pi<8$; in other words, $2\sqrt2<\pi<4$.