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.

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    $\begingroup$ Why do you suspect a closed form for the sum? $\endgroup$ – Did Oct 19 '14 at 15:58
  • $\begingroup$ @Did you mean it is not possible to find in closed form? $\endgroup$ – Arpit Kansal Oct 19 '14 at 16:00
  • $\begingroup$ 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? $\endgroup$ – Did Oct 19 '14 at 16:01
  • $\begingroup$ @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? $\endgroup$ – Arpit Kansal Oct 19 '14 at 16:05
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    $\begingroup$ Then why do you ask "find the sum"? This is misleading and, technically, makes your question not answerable. $\endgroup$ – Did Oct 19 '14 at 16:07

Hints for the convergence:

  • 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 $$\sum_{n=1}^\infty\frac{-1}{n(10^n-1)}.$$


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