Re: hot to write out this summation:

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.
If you settle for .0...1... you will have to clarify how .022332...23550... should be interpreted.
A harder question is what the above actually means, beyond just a way of describing (well-ordered) infinite strings.
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.
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:
  .((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 !!
Regards,
Mike.