Is the infinite sum of reciprocals powers of 2 a rational number?

Question

I am trying to prove that the sum $$\sum_{n \in J}^{\infty} 2^{-n}$$ is a rational number, where $J$ is some infinite subset of $\mathbb{N}$, but not equal $\mathbb{N}$. That is, I want to show that $1 + \frac{1}{4} + \frac{1}{32} + \cdots$ is a rational number. I tried to prove by induction that every partial sum is a rational number, which seem'd okay, but I know I can't generalize to say that if all partial sums are rational, then the infinity sum is rational. So I'm stuck... and would appreciate any help!

Answer 1

$\sum\limits_{n \in J} 2^{-n}$ is not always rational. For example, let $J=\{0!,1!,2!,3!,4!...\}$.

$\dfrac1{2^{0!}}+\dfrac1{2^{1!}}+\dfrac1{2^{2!}}+\dfrac1{2^{3!}}+\dfrac1{2^{4!}}+...$ is a Liouville number,

a non-repeating decimal, and thus irrational.