We have
[tex]0.6+0.66+0.666+0.6666[/tex]
Let's take 6 out from the series
[tex]6(0.1+0.11+0.111+0.1111+....[/tex]
Now multiply and divide by 9
[tex]\frac{6}{9}(0.9+0.99+0.999+0.9999+....[/tex]
[tex]\frac{6}{9}(1-0.1+1-0.01+1-0.001+1-0.0001+....
\\
\\\frac{6}{9}(1-10^{-1}+1-10^{-2}+1-10^{-3}+...[/tex]
all 1s add up to n
[tex]\frac{6}{9}(n-\frac{1}{10}\times \frac{1-(\frac{1}{10})^{n+1}}{1-\frac{1}{10}})[/tex]
