How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system? 
How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system ?

$0.\overline{101}=\underbrace{\dfrac{1}{2}+\dfrac{0}{4}+\dfrac{1}{8}}_{\dfrac{5}{8}}+\underbrace{\dfrac{1}{16}+\dfrac{0}{32}+\dfrac{1}{64}}_{\dfrac{5}{64}}+...$, 
then i get:
$\sum_{k\ge0}\dfrac{5}{8}\cdot (\dfrac{1}{8})^{k}=\dfrac{5}{7}$
but must it not be $(0.\overline{101})_2=\dfrac{101}{111}=(0.\overline{909})_{10}$
Thanks in advance

There is no such thing as a characeterwie substitution to obtain base ten periods from base two periods. Just thinkn of $\frac13=0.\overline 3_{10}=0.\overline{01}_2$ and especially, $\frac15=0.2_{10}=0.\overline{0011}_2$.
The actual analogy is that $$0.\overline{909}_{10}=\frac{909}{10^3-1}=\frac{909}{999}=\frac{101}{111}$$ and $$0.\overline{101}_2 = \frac{101_2}{2^3-1}=\frac 57.$$.
