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Paul
Posts:
507
Registered:
7/12/10


Re: is e < pi ?
Posted:
Jan 11, 2013 11:57 AM


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



