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Can this series be expressed in closed form, and if so, what is it? $$ \sum_{n=1}^\infty\frac{1}{9^{n+1}-1} $$

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I think there is a closed form for such things using the digamma function, if that qualifies as closed form for you. –  André Nicolas Sep 26 '12 at 23:27
@AndréNicolas well, I suppose it's better than nothing. How do you express it in terms of the digamma function? –  Navin Sep 27 '12 at 0:49
Erdos asked whether numbers of that sort are irrational. Peter Borwein settled that question in a paper in the Journal of Number Theory, volume 37 (1991), no. 3, pages 253–259. –  Gerry Myerson Sep 27 '12 at 0:51
@Navin: It was sort of a guess. Go to Maple, or Mathematica, or Alpha and it will tell you. –  André Nicolas Sep 27 '12 at 1:22
Lambert series may help as well as the q-digamma function –  Raymond Manzoni Sep 27 '12 at 19:45

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