I don't understand the last step:
When Archimedes did his calculuations he used a lower bound as well. Then he used the squeeze theorem to say that the value of Pi must be inbetween the two values. The above diagram only gives you an upper bound on the value of Pi.
@ Fernando haha that is an awsome observation!
Again this just shows that pi is less than for but bigger than 2.83
For a similar fallacy, let the broken line from (0, 0) to (1/2, 0) to (1/2, 1/2) to (1, 1/2) to (1, 1) be an "approximation" to the straight line from (0, 0) to (1, 1)- it has length 1/2+ 1/2+ 1/2= 2. A better approximation would be the broken line from (0, 0) to (1/4, 0) to (1/4, 1/4) to (1/2, 1/4) to (1/2, 1/2) to (3/4, 1/2) to (3/4, 3/4) to (1, 3/4) to (1, 1) for a total length of 8(1/4)= 2 again. In general, you have 2n line segments each of length 1/n. "In the limit" this appears to converge to the straight line, which has length , but the total length is always 2.
this is true. also on physicsfoums.com, and other places.
this is the nicest "non-technical" explanation i have seen:
when we approximate things in calculus, we always have a way of showing that the uncertainty, or error of our calculations goes to 0, in the limit.
in this case, the length of the approximating curve, the zig-zagging polygon, always stays the same, so the error is always 4-pi.
so, in the limit, all we know is that pi ≤ pi + 4-pi, which is true.