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Topic: e^x , Limits and Exponentiation in Non-Standard Reals.
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Posts: 134
Registered: 11/12/12
e^x , Limits and Exponentiation in Non-Standard Reals.
Posted: Feb 10, 2013 6:01 PM
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Hi, All:

I'm trying to see how to approach the exponential e^x , where we define:

e^x := lim_n-->oo ( 1+ x/n )^n (##)

Now, let's consider the NS Reals R* as equivalence classes in R^N (real-valued

sequences) by an ultrafilter. Can we meaningfully substitute the n above in

(## ) by some infinite hyperreal , say, y ? If so, does the result depend

on the choice of infinite hyperreal?

The addition and quotient in (##) can be done relatively straightforward:

Define 1:= (1,1,1,....,1,....) ; x=(x,x,x,....,x,.... ) , and to

simplify , y has no zeros (as terms in the sequence repping the class of y),

but , as a sequence y=(y1,y2,....), goes to infinity, i.e., yn-->.oo as

n-->oo .

Then (1+x/y)^y becomes:

(1+ x/y_1 , 1+x/y^2 ,.......,1+x/y^n ,............ )^y (%%%)

Now, do we exponentiate term-by-term, i.e., is the expression (%%%)

above equivalent to :

( (1+x/y_1)^y_1 , (1+ x/y_2)^y_2,......., (1+x/y_n)^y_n,......) ?

I'm just not clear on how to approach "non-standard infinity" , does that

mean selecting any infinite hyperreal?


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