For fun, I have been considering the number
$ \ell := \sum_{p} \frac{1}{2^p} $
It is clear that the sum converges and hence $\ell$ is finite. $\ell$ also has the binary expansion $ \ell = 0.01101010001\dots_2 $ with a $1$ in the $p^{th}$ place and zeroes elsewhere. I have also computed a few terms (and with the help of Wolfram Alpha, Plouffe's Inverter, and this link from Plouffe's Inverter) I have found that $\ell$ has the decimal expansion
$ \ell = .4146825098511116602481096221543077083657742381379169778682454144\dots. $ Based on the decimal expansion and the fact that $\ell$ can be well approximated by rationals, it seems exceedingly likely that $\ell$ is irrational. However, I have been unable to prove this.
Question: Can anyone provide a proof that $\ell$ is irrational?