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It is well known that harmonic series $$\sum_{n=1}^{\infty}\frac{1}{n}$$ diverges but in 1985 G. H. Behforooz proved that if we remove terms that have denominator that ends with $9$ series converges. To which constant does that series converge? What is special about numbers that end in $9$?

Jonas Meyer
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Adi Dani
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  • Could you please give a reference? I don't understand what the assertion is that Behforooz proved. – Jonas Meyer Feb 03 '13 at 00:05
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    I think you mean "if we remove all terms where the denominator has a $9$ in its base $10$ decimal expansion, then the series converges." The series most certainly still diverges if we remove only those numbers which end with a $9$. – Eric Naslund Feb 03 '13 at 00:09
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    The number $9$ is special. For example, cats have $9$ lives. But there is nothing special about $9$ for this problem. And it is not "ends in $9$", that does not affect divergence. – André Nicolas Feb 03 '13 at 00:20
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    See http://en.wikipedia.org/wiki/Kempner_series –  Feb 03 '13 at 00:48
  • @Byron. I think that link is very helpful. Thank you – Adi Dani Feb 03 '13 at 00:54
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    I found a reference to H (not G H) Behforooz in 1995 (not 1985): http://www.cut-the-knot.org/arithmetic/algebra/HarmonicSeries.shtml --- many other links arise from typing behforooz harmonic into the web. – Gerry Myerson Feb 03 '13 at 01:09
  • G Hossein Behforooz wrote an article called "Thinning out the harmonic series" that appeared in the October 1995 issue of Mathematics Magazine, the MAA publication. –  Feb 03 '13 at 01:33
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    @ByronSchmuland, please write up your insights as an answer. – vonbrand Feb 03 '13 at 02:37
  • @AndréNicolas Overhere, cats have 7 lives... – Felix Marin Jul 28 '14 at 23:57

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