A Nice Presentation of the Harmonic Series

Here is a nice video here about the Harmonic series. I would love for all my Pre-calc and calc students to see this video just to see the really nice proof by contradiction that the series is infinite. The paradox itself is just a bonus, and the fact that the squares add up to a value involving pi is a beautiful story that ought not to be popped out as an aside... sometimes we need to take the time to dig a little deeper into what Euler has given us... but a nice presentation all the same. Enjoy, but remember the adomonition that Borovik gave to his students,

" Today I said to the calculus students, "I know, you're looking at this series and you don't see what I'm warning you about. you look at it and you think, 'I trust this series. I would take candy from this series. I would get in a car with this series.' But I'm going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.

For more about the harmonic series, here and here and one more here, but surely there are more...use the search function.

A Rational Triangle Idea

Last week Sol at "Wild About Math" posted a link to the Wolfram Demo site related to a novel relationship that is not well known to students/teachers. For every rational number,q, on the open interval from 0 to 1, there is a Pythagorean Triple.

Actually the demo may oversell the idea a little, or perhaps I don't completely understand it, but I don't think it enumerates all the Pythagorean triples, but it is still a nice idea to introduce to students because many of the ideas can be proved with nothing beyond a good grasp of Alg II (for non-US folks, that's about a 14-16 year old students grasp of Algebra).

I had written about the relation back at the end of 2009. If you haven't read that one, and are not familiar with the idea, you might want to start there.

Anyway, Sol's post got me thinking again about how I would approach this idea with students. The first big idea is that if you pick a point on the y-axis between y=0 and y=1, with a rational y-coefficient, then the line through that point and the point (-1,0) will cross the unit circle in the first quadrant in a point (p,r) whose x and y coordinates are both rational. Since the point is on the unit circle, that means that p²+r² = 1. But since both p and r are rational, they can be written with a common denominator, Wolfram uses (a/c, b/c) and so (a/c)²+(b/c)²=1. If we multiply both sides of the expression by c², we get a²+b²= c² and so {a,b,c} is a Pythagorean triple.

So what might we ask our young charges to do with this? Well, first, do we believe that ANY rational on the positive y axis REALLY will produce a rational intersection on the unit circle? The proof requires no more than the solution of a quadratic. If the students are not up to challenging an open proof, have them start by confirming that some given rational y-intercepts will produce a rational intersection on the unit circle, and then go on to find the Pythagorean triple associated with that rational. For example, Wolfram's demo seems to associate the rational 1/2 with a (3,4,5) triangle... can they show why? Can you find the triple associated with 3/7?

I mentioned above that I didn't think this could produce ALL the Pythagorean triples... and we might want to ask our students if they think it could. One of my concerns about the Wolfram presentation is that the point (0, 1/3) is associated with the triple (8,6,10). Why not (4,3,5)??? how did they arrive at the non-primitive triple they chose, and not, for instance, (12,9,15) or ??? pick your favorite. I knew that they had used (3,4,5) in the first example (0, 1/2)...

In fact, would (9,12,15) show up later if I plowed through enough rational fractions? I suspected it wouldn't. First, the symmetry of the situation seemed to play against it. In simplest form, the x coordinate for any version of a (3,4,5) triple must be either 3/5 or 4/5; or at least so it seemed to me.

I think I would challenge my students to work backwards from a given triple and see if they could find the related rational y-intercept. Since we know (see my earlier blog) that the x-coordinate was (1-t²)/(1+t²), we could take any primitive triple, say (5,12,13) and set either the x value = 5/13 or 12/13 and get the two (and only two) values that would generate appropriate intersections on the unit circle. As we do this, we discover that the rational always turns out to be
where H is the hypotenuse of the triple, and L is the leg we chose to use for the x-coordinate ratio. For (5,12,13) we get the two rational values 2/3 (this is number 3 on the Wolfram demo) and 1/5 (number 7, but labeled (24, 10, 26) ) ... Bright students should be challenged to prove that any Primitive Pythagorean triple, will also be such that the ratio (H-L)/(H+L) give will produce a square of a rational number.

So maybe the Wolfram demo is just good enough (and just bad enough) to motivate our students, after all, they love exposing our flaws.

"It could easily be shown..." Probability and Pi and the Riemann Zeta Function

Beware of articles that begin, "It could easily be shown..." It is like arm wrestling a two year old, if you win, so what, and if you LOSE??? Yow....
I know this, and yet, I still proceed foolishly to read them. The one currently on my mind was a "Note on Pi" by R. Chartes in the March 1904 Philosophical Magazine, my current old document of choice. It pointed out that ICEBS "that if two numbers are written down at random, the probability that they will be prime to each other is 6/pi²."
Here it is from Wolfram Mathworld:

This is the reciprocal of the famous answer to the Basel problem evaluate by Euler.

The fact that the probability that two random numbers are relatively prime was equal to this value was discovered by M. Cesaro and J. J. Sylvester in the same year, 1883. Sylvester gives a proof in a footnote to a paper I found in his collected works (page 602)

The original image from Sylvester seems to have dissappeared into the abyss since 2009, so I share a proof I found on the Physics Harvard edu websight,

Ok, yeah, that is sort of easy, and I should have figured it out...The proof is easy to extend to the probability that three numbers are relatively prime is the reciprocal of the sum of the reciprocal of the cubes (if that seems hard to read, try to write it). More simply, the probability is the reciprocal of ζ(3)=

113+123+133+143+⋯

 Let  a,b and c be integers chosen at random

$<br>$

The probability that a, b, and c have no common divisor:

Pr(⊥(a,b,c))=1ζ(3)$Pr(\perp (𝑎,𝑏,𝑐))={\displaystyle \frac{1}{𝜁\left(3\right)}}$

where ζ$𝜁$ denotes the zeta function:

ζ(3)=113+123+133+143+⋯

the decimal value is approximately 
1ζ(3)

1ζ(3)=0⋅831907372580707…

 Strangely, the discovery (by Sylvester) is nested in work he was doing with Farey Fractions.  

If you haven't been exposed to Farey Fractions, a quick share from Wolfram's Mathworld

The Farey sequence  for any positive integer  is the set of irreducible rational numbers  with  and  arranged in increasing order. The first few are

			(1)
			(2)
			(3)
			(4)
			(5)

(OEIS A006842 and A006843). Except for , each  has an odd number of terms and the middle term is always 1/2.

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The image at the top shows a pair of coordinate axes with a point (x,y) painter black if GCF(m,n)=1, and white otherwise.

I think one of the really nice things that can be done with younger students studying common factors and slope (can I say in Alg I?) is to show them that the greatest common factor of m and n is the number of lattice points on the line from (0,0) to (m,n)....[not counting (0,0)] Here is a graph of the segments to (4,10) and (12,3)