Question

Find the sum of the following series up to $n$ terms: $.6+.66+.666+....$

Answer 1

Let, $S_n=.6+.66+.666+...tonterms=6[0.1+0.11+0.111+...tonterms]=\frac{6}{9}[0.9+0.99+0.999+...tonterms]=\frac{6}{9}[(1-\frac{1}{10})+(1-\frac{1}{{10}^2})+(1-\frac{1}{{10}^3})+...tonterms]=\frac{2}{3}\left[\right(1+1+...nterms)-\frac{1}{10}(1+\frac{1}{10}+\frac{1}{{10}^2}+...nterms)]=\frac{2}{3}[n-\frac{1}{10}\left(\frac{1-{\left(\frac{1}{10}\right)}^n}{1-\frac{1}{10}}\right)]=\frac{2}{3}n-\frac{2}{30}\times \frac{10}{9}(1-{10}^{-n})=\frac{2}{3}n-\frac{2}{27}(1-{10}^{-n})$