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