# Discuss convergence and find sum of the Series

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.

• Why do you suspect a closed form for the sum? – Did Oct 19 '14 at 15:58
• @Did you mean it is not possible to find in closed form? – Arpit Kansal Oct 19 '14 at 16:00
• I mean what I wrote (amazing, eh): why do you suspect a closed form for the sum, since you are asking (ordering, actually) for one? – Did Oct 19 '14 at 16:01
• @Did Actually I tried it for a long time but could not find any closed form so i was wondering whether is it possible or not? – Arpit Kansal Oct 19 '14 at 16:05
• Then why do you ask "find the sum"? This is misleading and, technically, makes your question not answerable. – Did Oct 19 '14 at 16:07

• For every $x$ in $(0,\frac1{10})$, $-2x\leqslant\log(1-x)\leqslant-x$.
• The series $\sum\limits_na^n$ converges absolutely for every $|a|\lt1$.
No reason to expect a closed form for the sum, an equivalent formula is $$\sum_{n=1}^\infty\frac{-1}{n(10^n-1)}.$$