Re: hot to write out this summation:

On 6/30/2024 3:44 PM, Mike Terry wrote:

On 30/06/2024 22:38, Chris M. Thomasson wrote:

r[0] = .01
r[1] = .0011
r[2] = .000111
r[3] = .00001111
r[4] = .0000011111
...
>
Now, think of a possible formula. Something like this shit:
>
;^) lol.
>
r[0] = .01
r[1] = .01 * 10^(-1) + .1 * 10^(-3) = .0011
r[2] = .0011 * 10^(-1) + .1 * 10^(-5) = .000111
r[3] = .000111 * 10^(-1) + .1 * 10^(-7) = .00001111
r[4] = .00001111 * 10^(-1) + .1 * 10^(-9) = .0000011111
r[5] = .0000011111 * 10^(-1) + .1 * 10^(-11) = .000000111111
...
>
Taken to infinity,

 .. to Infinity And Beyond !!  :)
 

it would be: .(0)(1) or something? How to properly write it:
>
.0...1...
>
?

 I'd go for .(0)(1)  specifying a point '.' then w (first infinite ordinal) '0's then w '1's.

I agree. .(0)(1) the first part is an infinity of zeros and the last part is an infinity of ones.

 If you settle for .0...1... you will have to clarify how .022332...23550... should be interpreted.

Yeah. It's more confusing that just .(0)(1)

A harder question is what the above actually means, beyond just a way of describing (well-ordered) infinite strings.

This might be missing your main point, but when zooming into some of my work, or even adjusting a parameter that alters the field, there are some big time differences between say:
.000111
and
.00001111
The differences show up in the final render result.
Not quite sure if that even relevant or not...
Humm.... Just thinking out loud here. Sorry! ;^o

E.g. you didn't put quotes around anything, making it look like you were describing regular real numbers, but regular real numbers have decimal notation with digits (after the decimal point) just at positions 1,2,...n,... with n < w.  So a "string" like .(0)(1) does not represent a real number. And "taken to infinity" /as a sequence of real numbers/ your sequence obviously converges to the real number 0.

Well, humm... Each iterate of .(0)(1) is a real number as in:
r[0] = .01
r[1] = .0011
r[2] = .000111
r[3] = .00001111
...
Each one is a real number, and each index is an unsigned integer? Fair enough? Mapping is fun. :^)

More generally you might search for a notation to represent "strings" of digits indexed by any ordinal, rather than just the simple ordinals w and 2*w in your example.  As a starter, what might the following designate:

Hummm... For some reason this is making me think of L-Systems.

  .((1)(2))
  .((12))
and what might be an example of a string with w^w digits?
 ..or why stick to ordinals?  Be brave!  To Infinity and Beyond !!

Thanks for your input Mike!  Thanks.