Puzzled about the output of my demo of a proof of The Euler Series
Richard D. Moores
rdmoores at gmail.com
Fri Aug 12 12:52:47 CEST 2011
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
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