|
|
e^x , Limits and Exponentiation in Non-Standard Reals.
Posted:
Feb 10, 2013 6:01 PM
|
|
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?
Thanks.
|
|
