1

In particular, I'm interested in the rationals that result from adding a finite sequence of consecutive integer powers of two. Has it been studied somewhere?

Update
This formalized example might clarify the point:
$$\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{} \frac{a}{b}=\sum_{x\in X} x,X \subset \mathscr P(\mathbb{R})\right\}$$ Can't provide concrete numeric examples so far.

nightcod3r
  • 365
  • Well, $\sum_{i=0}^n 2^i=2^{n+1}-1$ So $\sum_{i=m}^n 2^i=2^{n+1}-1-(2^{m}-1)=2^{n+1}-2^{m}$. Is that what you meant? – lulu Mar 14 '16 at 12:26
  • I mean fractional numbers, whose property is that each one represents the summation of a sequence of numbers (integer powers, for example). Are there subsets that meet this property described in the literature? – nightcod3r Mar 14 '16 at 12:33
  • 1
    I don't understand. Can you edit your question to give some examples of what you mean? You say "rationals that result from adding a finite sequence of consecutive integer powers of two." Those can only be integers. – lulu Mar 14 '16 at 12:35
  • Note: even if you meant to include negative integer powers, the geometric series calculation sketched in an earlier comment let's you compute such expressions exactly. But, really, some concrete numerical examples of what you are interested in would clarify things. – lulu Mar 14 '16 at 12:44
  • Clarification added. – nightcod3r Mar 14 '16 at 13:11
  • If by $\mathscr P$ you mean "powers" then every element of $\mathscr P(\mathbb N)$ is an integer so all those elements are integers. Can you give one single example of a non-integral fraction that arises in the way you want? Also, if $\mathscr P(\mathbb N)$ includes first powers then every integer arises this way. For that matter, if it includes squares, then every integer arises this way. – lulu Mar 14 '16 at 13:44
  • It represents the power set, such that $X$ is a subset of naturals (for example). $3/4=2^{-1}+2^{-2}$, $7/16=2^{-2}+2^{-3}+2^{-4}$, they are examples of rationals (fractional numbers) that result of adding a finite series of integer powers of two. – nightcod3r Mar 14 '16 at 14:32
  • Ok...your definition specifically referred to powers of natural numbers, negative integers are not "natural numbers". And, as I said, if you want consecutive powers of $2$, positive or negative, it is easy to evaluate them using geometric series. Sorry, I'm not understanding what you want. Good luck! – lulu Mar 14 '16 at 14:35
  • The question is more general, it refers to any sequence of number, in the formal example I propose to add naturals, in such a case the rational would be integers, of course. The example in this thread includes integer powers of two, just another possible case. But will edit the example in the question to be more general. I know it might be intricate, but thanks for your help. – nightcod3r Mar 14 '16 at 14:43

0 Answers0