Question

It is well known that the harmonic series, 1/1 + 1/2 + 1/3 + 1/4 + ... , diverges. Consider a depleted harmonic series; see below; which contains only terms whose denominator does not contain a 9. (In decimal representation.) Does this series diverge or converge?

S = 1/1 + 1/2 + ... + 1/8 + 1/10 + ... + 1/18 + 1/20 + ... + 1/88 + 1/100 + 1/101 + ... It is well known that the harmonic series, 1/1 + 1/2 + 1/3 + 1/4 + ... , diverges. Consider a depleted harmonic series; see below; which contains only terms whose denominator does not contain a 9. (In decimal representation.) Does this series diverge or converge?

S = 1/1 + 1/2 + ... + 1/8 + 1/10 + ... + 1/18 + 1/20 + ... + 1/88 + 1/100 + 1/101 + ... @Mathematics

Answer 1

I think it converges, and that there is a proof.

Answer 2

http://mathworld.wolfram.com/KempnerSeries.html

Answer 3

hint: Group terms according to the number of digits in their denominator.

Answer 4

http://www.qbyte.org/puzzles/p072s.html

Answer 5

agree with ag.

If the Kempner converges, with a term less, that series which contains only terms whose denominator does not contain a 9, will converge.

Answer 6

The depleted harmonic series, which contains only terms whose denominator does not contain a 9, does not diverge. It can be shown to converge to a value less than 80.