3
$\begingroup$

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 $9$?

$\endgroup$
  • 8
    $\begingroup$ 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 $9$. $\endgroup$ – Eric Naslund Feb 3 '13 at 0:09
  • 3
    $\begingroup$ 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. $\endgroup$ – André Nicolas Feb 3 '13 at 0:20
  • 5
    $\begingroup$ See en.wikipedia.org/wiki/Kempner_series $\endgroup$ – user940 Feb 3 '13 at 0:48
  • 1
    $\begingroup$ I found a reference to H (not G H) Behforooz in 1995 (not 1985): cut-the-knot.org/arithmetic/algebra/HarmonicSeries.shtml --- many other links arise from typing behforooz harmonic into the web. $\endgroup$ – Gerry Myerson Feb 3 '13 at 1:09
  • 1
    $\begingroup$ @ByronSchmuland, please write up your insights as an answer. $\endgroup$ – vonbrand Feb 3 '13 at 2:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.