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


Re: The Many Roles of e
Posted:
Apr 21, 2013 1:37 PM


>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 y2=z, we can rewrite your >expression as >[(2^x+2)/(2^x2)]^(2^x)={(y+2)/(y2)]^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)]/[22^(x)]}^[2^(x2)] >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



