[seqfan] found a new sequence, not sure if interesting, and how to properly submit
Jerry V Polfer
polfer.jerry at gmail.com
Sun Jan 22 13:51:52 CET 2017
Since i'm new on this list, i'd first like to say hello to everyone.
I'm not really mathematically trained (except the bits i learned in my
mechanical engeneering studies), after leaving school and hating
mathematics, i now found some newly developed interest in it, mainly by
numberphile (and other) videos on YouTube.
Inspired by such a video[1] called "Riemann's paradox: pi = infinity minus
infinity" which was about Riemann series theorem,i had the idea for a
sequence.
According to this theorem, you can get that series to diverge to any number
you want, by rearranging the terms. In the video it is done by first adding
up as many positive terms as you need to get a sum >= n, then subtracting
as many negative ones as you need to bring the total sum to a number < n,
then continuing with positive terms and so on.
So how many positive terms of the infinite series "1 - 1/2 + 1/3 - 1/4 +
1/5 - 1/6 + ..." do we need to add up to arrive at a sum >= n.
n terms
1 1 (1=1)
2 8 (1+1/3+1/5+1/7+1/9+1/11+1/13+1/15=2.0218004...)
3 57 (sum=3.0032870...)
4 419 (sum=4.0006905...)
5 3092 (sum=5.0000417...)
6 22846 <22%20846> (sum=6.0000206...)
7 168804 (sum=7.0000017...)
8 1247298 (sum=8.00000009783...)
9 9216354 <92%2016%2035%204> (sum=9.00000004932...)
10 68100151 (sum=10.0000000072374...)
11 503195829 <50%2031%2095%20829> (sum=11.0000000001968...)
12 3718142208 <37%2018%2014%202208> (sum=12.0000000000281...)
This numbers i did find using this script, a rather brute force method:
http://pastebin.com/VaCXhfT6
Interestingly, at least for me, the next number of terms can roughly be
predicted:
Number of terms for n = numbers of terms for n-1 * e^2
Now i don't know if this sequence would be of any interest for the OEIS,
and secondly how to properly submit, since i don't know any
mathematica/maple code, and miss the knowledge of proper terms to describe
it and formulate it in a formula.
Can i submit it with nearly next to none extra information, and hope others
clean up my mess, or should i refrain from submitting?
Kind regards and sorry for the wall of text,
Jerry Polfer
[1] https://www.youtube.com/watch?v=-EtHF5ND3_s
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