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dan73
Posts:
468
From:
ct
Registered:
6/14/08


Re: The Many Roles of e
Posted:
Apr 20, 2013 10:05 PM


> I am collecting results for a paper or monograph on > the appearance of the constant e in diverse aeas of > mathematics, and am asking anyone with examples to > let me know about them so they may be included. Here > are a few, from wellknown to rather exotic: > > 1: The classical definition found in attempts to > differentiate the logarithm function can also be > computed and shown to exist using the "area" > definition. > 2: Probablility (e.g., "If a diner at random picks a > hat at random, ...") and statistics (e.g., limits of > binomial distributions are Poisson distributions). > 3: It is a "factorially stable" irrational number, > meaning that the sequence {n!e[n!e]} has a limit > (it's 0). > 4: Here's my favorite: The "exponential tower" > x^(x^(x^(...))) converges on the interval [e^(e), > e^(1/e)] and has range [1/e, e]. > > Of course, there are many other expressions (limits, > definite integrals, ...) that perhaps surprisingly be > expressed in terms of e, and I would like to "collect > them all". At present, the interest is only in real > analysis, but any complex results are also solicited. > Wherever results are used, credit will be given to > o the the contributor. Thanks for your attention  I > look forward to a lively dialogue and more surprises > about this remarkable number.
My favorite closed form for (e) was created by Harlen Brothers and associates.
[(2+2^x)/(22^x)]^2^x
If x =100
Correct too 61 digits of exponential (e)
The larger the x the more decimal digits match (e) This, I believe, is the best and simplest equation to calculate (e)
A slight twist to the above equation ties pi with (e) on a convergence too (4).
Where n = pi or a multiple of pi.
4~(in(( 2+2^n)/(2^n 2)^2^n))
When n=pi =4.07067... """""n=2pi =4.000879... """""n=3pi =4.000011290... """""n=4pi =4.000000144973... """""n=5pi =4.000000001861... ... """""n=10pi =4.00000000000000000064972... ... """""n=100pi =4.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003839495388522987805741...
It appears that each multiple of pi adds almost 2 zeros to the decimal expansion of (4)
Dan



