Valentin's recent challenge involving the harmonic series caused me to run across a related series:

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

and a fascinating modification to it.

If you delete from the harmonic series of primes those primes that contain each of the digits 0,1,2,3,4,5,6,7,8,9 at least once, the series converges.

A paper providing a proof of this was published in Mathematics Magazine in October, 1995.

But, the author didn't provide the value of the limit.

Can we find it with our calculators?

Lets call harmonic(x) the harmonic series with digit x missing, primes(x) the 1/primes series with digit x missing. Then we know primes(x) also converges since primes(x) < harmonic(x) which converges.

Your series is primes(0-9) = 1/primes striking out those terms missing at least one of 0-9. I think this is what you mean by, delete those containing “each” of 0-9. Ie those than contain all digits are removed. Otherwise if you mean any of 0-9 then clearly this is all terms!

So primes(x) is part of primes(0-9), since x is missing and primes(0-9) < primes(0) + primes(1) + .. + primes(9), since the latter counts some terms twice, but the latter has a value so the former must converge. So there’s a convergence proof and an upper bound.

This also means harmonic(0-9) converges and could be calculated by egan’s program. We also have primes(0-9) < harmonic(0-9) too.

Quote:

Your series is primes(0-9) = 1/primes striking out those terms missing at least one of 0-9. I think this is what you mean by, delete those containing “each” of 0-9. Ie those than contain all digits are removed. Otherwise if you mean any of 0-9 then clearly this is all terms!

"striking out those terms missing at least one of 0-9" is not the same.

For example, your description would strike out 102345789 because it is missing the digit 6.

My description would not strike it out because it doesn't contain each of the digits 0,1,2,3,4,5,6,7,8,9 at least once; it doesn't contain the digit 6 at least once.

i see.

i've noticed that my argument applies to your definition and not to mine! my previous definition was wrong. in fact the series made by striking out those missing one of 0-9 will be the series of those containing all of 0-9 which will be nearly all terms (after a while).

your series is then the opposite. you are striking about those containing all of 0-9 so the series primes(0-9) is 1/primes with at least one of 0-9 missing.

so then, primes(x) is part of primes(0-9) since if x is missing, it can't be struck off.

and then primes(0-9) <= primes(0) + primes(1) + .. + primes(9) as before.