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Topic: The Many Roles of e
Replies: 8   Last Post: Apr 23, 2013 8:39 AM

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dan73

Posts: 465
From: ct
Registered: 6/14/08
Re: The Many Roles of e
Posted: Apr 20, 2013 10:05 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

> 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 well-known 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)/(2-2^-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



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