Which brings me to the question: is the number 1.212212221... rational or irrational. The answers to the first questions I would think would help answer the previous question. If those answers are not sufficient enough to determine if this number is rational or irrational, please let me know how one would go about proving the rationality/irrationality for 1.212212221... If it helps, I came up with an exact representation of this number using summation notation, which is shown in the attached JPEG.
If 1.212212221... is irrational (which is where I'm leaning towards) then is it considered to be a special type of irrational number with interesting properties? Notice that in the JPEG, summation in the exponents is used, a use of summation that I'm not very familiar with.
Thanks in advanced for considering my questions.