Show that the series $\sum_{n=1}^\infty \ln(1-\frac{{1}}{10^n})$ converges and find the sum in closed form if it is possible.
Try:Clearly given series converges because if $0<a_n<1$ then $\sum_{n=1}^\infty \ln(1-a_n)$ converges iff $\sum_{n=1}^\infty a_n $ converges. Give some hint for finding the sum of series.