Puzzled about the output of my demo of a proof of The Euler Series

[Richard D. Moores]

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.

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