1/1+1/3+1/6...=2 (cont.)

Can I run this by some math people, at the risk of feeling like a
clown with a wizards hat on? (It's related to above thread/topic. I
was really impressed by Gene's recognizing that the series converged
on 2, and Robert's proof of it.) I have no skill in math (though
the numbers relate to chords I compose with), so this is probably
just hokus pocus number mysticism...but just in case there's
something to it.

I recall the importance of odd numbers (and primes) in 'lattice
theory' etc discussed here, and noticed that counting numbers and odd
numbers give triangular numbers in the following manner: 3x1=3,
3x2=6; 5x2=10, 5x3=15; 7x3=21, 7x4=28; et cetera.

Since the above thread was about the recipricals of counting numbers
(the harmonic series) and the recipricals of triangle numbers, I
thought I would mention it. (And naturally, the fact that each odd
number is repeated twice in the progression, reflects a connection to
the first series converging on 2...just kidding...)

Kelly

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> I recall the importance of odd numbers (and primes) in 'lattice
> theory' etc discussed here, and noticed that counting numbers and odd
> numbers give triangular numbers in the following manner: 3x1=3,
> 3x2=6; 5x2=10, 5x3=15; 7x3=21, 7x4=28; et cetera.

The formula for the nth triangular number (which you can use the same
method of indiction to prove as Robert used) is n(n+1)/2; since either
n or n+1 is an even number, you get the pattern you observed.

The formula for the nth tetrahedronal number is n(n+1)(n+2)/6; you can
amuse yourself by finding what the sum of its reciprocals comes to.
It's strange but true that these infinite series are easy to sum, but
reciprocals of powers lead into deeper waters. The values for even
powers are known and known to be transcendental, but mathematicians
were much surprised when Apery proved that the sum of the reciprocals
of the cubes is an irrational number--something everyone assumed was
true, but no one expected a proof for.

Hi Kelly,

> I recall the importance of odd numbers (and primes) in 'lattice
> theory' etc discussed here, and noticed that counting numbers and odd
> numbers give triangular numbers in the following manner: 3x1=3,
> 3x2=6; 5x2=10, 5x3=15; 7x3=21, 7x4=28; et cetera.

Yes that's the formula n(n+1)/2

3*4/2 = 6, 4*5/2 = 10, 5*6/2 = 15, 6*7/2 = 21 ...

It is easy to prove too:

To get the induction started, it's true for n = 2:

1 + 2 = 2(2+1)/2

So, now we want to get from any arbitrary n to n+1, so
from
1 + 2 + ... + n = n(n+1)/2

add n+1 and you get
1 + 2 + ... + n + n+1 = n(n+1)/2 + n + 1
= (n+1) (n+2) / 2

So from the result for 2, you can get to the result
for 3, 4, ... and so as far as you like so it
is true for all n.

Or you can see it geometrically:

* * -
* - -

1 + 2 is half of a 2 by 3 rectangle

* * * -
* * - -
* - - -

1 + 2 + 3 is half of a 3 by 4 rectangle,

and so on.

A nice related result is that the sum of the first n odd numbers is n squared

1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
etc.

which you can understand geometrically by contemplating this diagram:

- * @ x % ~
* * @ x % ~
@ @ @ x % ~
x x x x % ~
% % % % % ~
~ ~ ~ ~ ~ ~

Robert

hi Gene

> reciprocals of powers lead into deeper waters. The values for even
> powers are known and known to be transcendental, but mathematicians
> were much surprised when Apery proved that the sum of the
reciprocals of the cubes is an irrational number--something everyone
assumed was true, but no one expected a proof for.>>

very interesting....thank you for pointing this out. (Reminds me of
passages -- about cubic numbers -- which I read in "Fermat's
Enigma"). Also nifty is the connection (from Euler?) between the
running total of the doubling series and perfect numbers...since you
mentioned powers --and since one of my favorite chords, c-f#-b, sums
to 28/15! (7/5x4/3)

--Kelly