This is an imaginary exercise that my math teacher gave me to meditate:
For any real number $(*)$ $r=\overline{r_0.r_1r_2…}$, expressed using $10$-base decimals, define the following sum:
$$S(r):=\sum_{i=1}^{\infty}(-1)^{i-1}r_i=r_1-r_2+r_3-r_4+…$$
Also, the partial sums are: $S_k(r):=\sum_{i=1}^k(-1)^{i-1}r_i$, for every natural number $k$.
We also define three classes/types of real numbers, based on this sum (below I construct some examples):
$1)$ $F$-type numbers, for which $S(r)$ exists and is finite, for example, $S(200.74)=3$, or any rational number with a finite number of nonzero decimals in $(*)$, like integers or $p/q$, with $p,q$ coprime and $q=2^a5^b$.
$2)$ $I$-type numbers, for which $S(r)$ exists and is infinite, for example, $S(0.(01))=-\infty, S(0.1010001000001…)=+\infty$. So, in this class, there are both rational and irrational numbers.
$3)$ $N$-type numbers, for which $S(r)$ does not exist, for example, $S(0.(1)), S(0.(1265)).$
!! The sum $S(0.(9))$ is not considered (although $0.(9)=1$), so for integers, we take all decimals after the floating point to be zero, ensuring that the $(*)$ representation is unique. As mentioned before, $\forall x \in \mathbb{Z}$ is $F$-type, with $S(x)=0$, so we are done with this.
I think we can fully classify in this way all rational numbers. Correct me if I am wrong.
My problem is whether we can fully classify the real numbers in this manner or not, and I am thinking about relevant irrational numbers like $\sqrt{p}$, with $p$ prime, $\pi$ or $e$. Or maybe this question is a topic similar to ,,$e+\pi$ is rational or not'', or other rationality uncertainties?
So $\mathbb{R}-\mathbb{Q}$ is my case of interest. For some irrational numbers, it is almost impossible to control their digits, but maybe for a certain subset, we can hope to precisely declare the $F/N/I$-type of the elements of it.
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If you encountered/heard something similar to this sum defined above, I am more than happy to find some articles/references to this. Maybe this sum was studied before by some great minds, but I can't check this.
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Also, I would be great if you could show me a graph of the partial sums $S_k(r)$ for some irrational numbers, the greater $k$, the better, just to see how it works/behaves. The most interesting would be $\pi$, of course.
Thank you for your time and effort!
P.S. I am not expecting to receive a positive answer, but rather a suggestion/opinion about this situation.
IMPORTANT EDIT: The problem is still open, although I have accepted the answer given (being the most relevant). Also, the afferent comments to this question are not negligible.
Best Answer
Except for terminating decimals, $r_i$ does not go to zero, so the series $S(r)$ does not converge. As noted, in some cases we have $S(r) = \pm\infty$.
However, we do have Cesaro convergence: for almost all reals $r$ $$\lim_{k\to\infty}\frac{1}{k}S_k(r) = 0.$$ To prove this note that $r_{2i-1}-r_{2i}$ are IID mean zero with respect to Lebesgue measure on $[0,1]$, and apply the strong law of large numbers.