lean_workbook_plus_34622

Let $x=\sum_{k=1}^{\infty }a_{k}3^{-k}, \ a_{k}=0,1,2,$ then $f(x)=\sum_{k}b_{k}2^{-k}$ ,\nwere $b_{k}=a_{k}/2$ if $a_{i}=0 \ or \ 2,i\le k$ , $b_{k}=1$ if $a_{k}=1,a_{i}\not =1,i<k$ and if $a_{k}=1$ , then $b_{i}=0 \ \forall i>k$ ..\nIt is easy to proof by induction.\nBut I don't calculate $x=\frac{18}{1991}=\frac{18*(3^{90}-1)/1991}{3^{90}-1}$ . These representation give expressions $a_{k}\ \forall k$ .\nBecause $a_{1}=a_{2}=a_{3}=a_{4}=0=a_{6},a_{5}=2,a_{7}=1$ we get $f(\frac{18}{1991}=2^{-5}+2^{-7}=\frac{5}{128}.$