Comments on: arXival mysteries

By: Jim Ward

Jim Ward — Mon, 14 Jul 2008 16:20:22 +0000

The Baillie and Thomas Schmelzer article point to Nick Trefethen’s TEN DIGIT ALGORITHMS - “Ten digits, five seconds, and just one page”

http://www.comlab.ox.ac.uk/nick.trefethen/tda.html

The algorithms might make a fun book.

By: Robert Baillie

Robert Baillie — Fri, 11 Jul 2008 10:46:59 +0000

to respond to craid kaplan… for kempner series: given a sum T, one can use trial and error to find a set (say, S) of digit strings such that the kempner sum of 1/n where n has no digits in S, is near T. i don’t know if that’s possible for irwin series, since it is not yet known how to compute the sum of 1/n where n has, say, one occurrence of a multi-digit string like “314″.

here’s another interesting point. let K = sum of kempner’s “no 9″ series:
K = 1/1 + 1/2 + … + 1/8 + 1/10 + … ~ 22.92067…
let I1 = sum of 1/n where n has one 9:
I1 = 1/9 + 1/19 + 1/29 + … .
the abstract says that I1 > K, even though the I1 series starts out being much smaller than the K series. but that’s also true for I2 = sum of 1/n where n has 2 nines:
I2 = 1/99 + 1/199 + 1/299 + … + 1/899 + 1/909 + … ~ 23.02604… > K.
the same is true for, say, 10 nines:
I10 = 1/9999999999 + 1/19999999999 + … ~ 23.02585092994045684022 > K.

the paper doesn’t prove that the sum is > K for an arbitrary number of nines, but the computational results suggest that for a large number of nines, the sum is close to 10*ln(10) ~ 23.02585092994045684018 .

By: Flocon

Flocon — Thu, 03 Jul 2008 06:12:41 +0000

In his book “Gamma: Exploring Euler’s Constant” Julian Havil devotes a few pages to the Kempner series. The book is worth checking out if you’re interested in the harmonic — and related — series.

By: Jack

Jack — Wed, 02 Jul 2008 21:20:40 +0000

My college analysis prof gave this series as an example. When you look at the first few dozen terms after removing from the harmonic series the terms having a 9 in them, it doesn’t seem like you’re omitting enough terms to bring the series into convergence.

When he began to explain why the series unintuitively converges, he opened with “Most integers have more than a billion digits” which was at the same time entertaining and illuminating.

By: Craig Kaplan

Craig Kaplan — Wed, 02 Jul 2008 16:51:58 +0000

Could you submit a dummy paper, obtain its reference number, and then update the paper in such a way that it incorportates the number? If you had a magic formula that, given a string of digits, produced a description of the form “the sum of 1/n where n has at most k occurrences of the digit d” that contained those digits, you’d be all set…