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Topic: is e < pi ?
Replies: 4   Last Post: Mar 9, 2013 8:18 AM

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Paul

Posts: 393
Registered: 7/12/10
Re: is e < pi ?
Posted: Jan 11, 2013 11:57 AM
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On Friday, January 11, 2013 4:10:28 PM UTC, rodp...@gmail.com wrote:
> An obvious answer. But can you prove it ?
>


1 + 1 + 1/2! + 1/3! is bounded by 1 + 1 + 1/2 + 1/(2^2) + 1/(2^3) ... = 3
So e < 3.

Unlike e, pi is defined differently by different authors. Any definition should enable the conclusion that pi > 3.

That's a bit vague. It can be shown that pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 etc.
Expand that series to about 10^6 terms using your favourite software. Use a finite expansion that gets chopped off before a positive term. That way you get a lower bound. This lower bound is > 3 . So pi > 3.

Paul Epstein



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