Consider the following number:
$\begin{align} R &= \frac{1}{9}\sum^\infty_{n=1} 10^{-\frac{n\left(n+1\right)}{2}}\left(10^n-1\right)\left(n\left(\operatorname{mod}10\right)\right) \\ &= 0.1223334444555556666667777777888888889999999990000000000111111111112222222\\&222223333333333333444444444444445555555555555556666666666666666777777777777777\\&77888888888888888888999999999999999999900000000000000000000\cdots \end{align}$
which is formed by concatenating $n$ copies of $n\left(\operatorname{mod}10\right)$ after $0$.
The long sequences of repeating digits allow better and better rational approximations as the lengths of repeating digit blocks grow. Can this be a basis to prove this number transcendental?
from Hot Weekly Questions - Mathematics Stack Exchange
Post a Comment