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

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  Puzzled about the output of my demo of a proof of The Euler Series "Richard D. Moores" <rdmoores@gmail.com> - 2011-08-10 16:57 -0700
    Re: Puzzled about the output of my demo of a proof of The Euler Series casevh <casevh@gmail.com> - 2011-08-10 22:44 -0700

#11158 — Puzzled about the output of my demo of a proof of The Euler Series

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-compute-sum-n-1-infty-frac1n2>.
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

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#11186

On Aug 10, 4:57 pm, "Richard D. Moores" <rdmoo...@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

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