For the first time in my 7 years of using Gmail, I accidentally deleted my original post and it's one reply by casevh. I found both in the list archive, and with this post both quote casevh's reply and answer it. Sorry about my screw up.
On Aug 10, 4:57 pm, "Richard D. Moores" <rdmoo... at gmail.com> wrote: > I saw an interesting proof of the limit of The Euler Series on > math.stackexchange.com at > <http://math.stackexchange.com/questions/8337/different-methods-to-com...>. > Scroll down to Hans Lundmark's post. > > I thought I'd try to see this "pinching down" on the limit of pi**2/6. > See my attempt, and output for n = 150 at > <http://pastebin.com/pvznFWsT>. What puzzles me is that > upper_bound_partial_sum (lines 39 and 60) is always smaller than the > limit. It should be greater than the limit, right? If not, no pinching > between upper_bound_partial_sum and lower_bound_partial_sum. > > I've checked and double-checked the computation, but can't figure out > what's wrong. > > Thanks, > > Dick Moores The math is correct. The proof only asserts that sum(1/k^2) is between the upper and lower partial sums. The upper and lower partial sums both converge to pi^2/6 from below and since the sum(1/k^2) is between the two partial sums, it must also converge to pi^2/6. Try calculating sum(1/k^2) for k in range(1, 2**n) and compare that with the upper and lower sums. I verified it with several values up to n=20. casevh ===================Dick Moores' reply=================== Thank you! I had missed the 2^n -1 on the top of the sigma (see my image of the inequality expression at <http://www.rcblue.com/images/PinchingForEuler.jpg>. So I rewrote the script and now it does what I intended -- show the pinching down on sum(1/k^2) by the upper sums and the lower sums for successively larger n. See the new script at <http://pastebin.com/PGXx7raq>. Dick -- http://mail.python.org/mailman/listinfo/python-list
