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dan73
Posts:
467
From:
ct
Registered:
6/14/08
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Re: The Many Roles of e
Posted:
Apr 21, 2013 1:37 PM
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>Hi Dan,
>This is pretty neat, but I think your expression >converges to e^4 as x -> infinity. Perhaps I am >misreading something, but here is what led to this >perception.
Yes as x -->oo the convergence to e^4 -->oo
When x =100 the correct decimal expansion of e is about
188 digits.
>If we let 2^x=y and y-2=z, we can rewrite your >expression as
>[(2^x+2)/(2^x-2)]^(2^x)={(y+2)/(y-2)]^y=[(z+4)/z]^(z+2)
>which approaches e^4 as z -> infinity. Continuing as if >the observation is correct, I think the expression
>{[2+2^(-x)]/[2-2^(-x)]}^[2^(x-2)]
>converges to e.
>Your second observation is also interesting - that the >convergence would be even more rapid if 2 is replaced by >a larger integer.
>I'm still absorbing your statements about the >relationship of e to pi. So e^(i pi)+1=0 is not the only >one!!
There are many more relationships of pi and e that are
much more complex then the simple one that I show.
>Cheers,
>Will
I just try different things like stealing Harlens'
equation and plugging in my own value for (n)=
pi or a multiple of pi.
When messing with math you never know what new
thing you might come up with.
This may not be new but expressed differently.
Have a fun search!
Dan
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