[math-fun] series convergence problem
I guess the answer to this is a known "folk theorem" but have not seen it explicitly stated before PUZZLE: Given a finite-dimensional vector V=(v0,v1,v2,...,vk) define the function F(n) via F(n) = n^v0 * ln(n)^v1 * lnln(n)^v2 * lnlnln(n)^v3 * ... Then: for which vectors V is it true that some N>1 exists such that the series sum(n>N) 1/F(n) converges? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
About 35 years ago, either Murray Klamkin or Andrew Gleason mentioned this problem to myself and some other students, and said that you look at the first entry v_i that's unequal to 1. If it's greater than 1, the proposition is true; if it's less than or equal to 1 (or if all the entries equal 1), then it's false. I got the impression that this might have been a Monthly problem. Jim Propp On Tuesday, July 23, 2013, Warren D Smith <warren.wds@gmail.com> wrote:
I guess the answer to this is a known "folk theorem" but have not seen it explicitly stated before
PUZZLE: Given a finite-dimensional vector V=(v0,v1,v2,...,vk) define the function F(n) via F(n) = n^v0 * ln(n)^v1 * lnln(n)^v2 * lnlnln(n)^v3 * ...
Then: for which vectors V is it true that some N>1 exists such that the series
sum(n>N) 1/F(n)
converges?
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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