For Just the Proof of the divergence of the Harmonic Series of Primes
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Just The Proof
Story Background and History
About 6 weeks ago I ran across the Harmonic Series in one of my books. It wasn’t the first time I’d seen it but… anyway…
The Harmonic Series goes like this…
Simple and straightforward, right? But what I found really interesting is a Proof that the series “diverges.” I.e. that the sum of the terms goes to infinity as n gets larger and larger (as n goes to infinity). The most interesting, simple, and elegant Proof of the divergence of the Harmonic Series is the one by Nicole d’Oresme (ca. 1323-1382),
Oresme’s proof groups the harmonic terms by taking 2, 4, 8, 16, … terms (after the first two) and noting that each such block has a sum larger than 1/2,
and since an infinite sum of halves diverges, so does the harmonic series diverge. This proof is quintessentially simple and elegant. That said, here’s my story about how Oresme’s proof led me to search for another SIMPLE PROOF except that would be a SIMPLE PROOF of the divergence of The Harmonic Series of Primes.
Lock The Gates!
I presumed there would be a similar well-known series, except for Primes. So I googled “Harmonic Series Primes” and voila!! We find the Harmonic Series of Primes. Here are a couple of links:
So here is what the Harmonic Series of Primes (HSOP) looks like…
All the way out to PN (1/2 + 1/3 + 1/5 + … 1/PN)
Simple. Straightforward. And I quickly learned that this series also diverged (the sum of the terms of the HSOP goes to infinity). I could find various proofs of this but they were complicated and hard to follow… at least for me. They weren’t nearly as simple as Oresme’s proof for the Harmonic Series. Now that doesn’t mean a very simple proof of the divergence of the HSOP did not exist… Only that I did not find one. If you find one the PLEASE let me know.
So that became my next project; a quest as it were to develop a SIMPLE proof.
The “quest” lasted about 6 weeks with ruminating about this while watching TMZ or while insomniating in bed at 3 in the morning listening to Art Bell or George Noory or the Tappet Brothers. I tried various approaches to the problem (multiple times and multiple variations for each approach)… a lot writing on white boards… but I didn’t get anywhere. It was frustrating.
In retrospect, as we’ll see shortly, I was hung up on the details of one particular part of one formula as opposed to what that part of that formula was, in general, trying to tell me!! Why it took weeks to see it is beyond me. However, there was one saving grace and it was that I intuitively felt that the Prime Number Theorem (PNT) would play a major (largest) role for a variety of reasons not the least of which is that it’s simple and elegant and packed full of information in a very small space! That intuition would turn out to be very important!
All that said, here’s how it goes. Let’s start with the Prime Number Theorem (PNT).
The PNT tells us “approximately” how many Primes there are between 0 and some integer N. It’s a formula. And the formula is:
π(N) ~ N / ln(N),
where π(N) is the prime-counting function
and ln(N) is the natural logarithm of N.
Note that “ ~ ” means “approximates closely (asymptotically).” For example, how many Primes are there between 0 and 1 million?
π(N) ~ N / ln(N)
so for 1 million it’s…
π(1,000,000) ~ 1,000,000 / ln(1,000,000)
So that is point 1. That is, The PNT tells us that the count of Primes between 0 and N is
To compute the HSOP out to any Prime PN we can calculate an “average” value for the terms by dividing the sum of the terms by the number of the terms. For example, let’s take the HSOP out just 4 terms:
1/2 + 1/3 + 1/5 + 1/7 = (105 + 70 + 42 + 30) / 210 = 247/210 = 1.1762
and if we divide 1.1762 (the sum of the terms) by 4 (the number of terms) , we get .2940476… which is the “average value” of the terms.
We can then multiply the average value of the terms by the count of the number of terms to obtain the sum of the terms (i.e. to obtain the value of the series)!!
.2940476 x 4 = 1.1762
Avg x Cnt = SumOfSeriesTerms
Think of it this way…
(TheSumOfTheSeriesTerms divided by TheCountOfTerms) = TheAverageValueOfTheTerms
or… Series / Cnt = Avg
which leads to
Series = Avg x Cnt and earlier we said that Avg x Cnt = Series so… Series = Series
and that seemed trivial and meaningless so I would follow some bunny trail and look for a different approach.
And that is what I was getting hung up on.
But it’s the wrong way to think of it!
Instead we need to think of it this way…
If we take the Harmonic Series of Primes out to the Nth prime then there is a number (a value) that is the “Average Value” of the terms of the series (call it AVG)
It turns out that the value of AVG is somewhere between ½ and PN/2 but I’ll leave that to the reader to ponder if they want to.
But we don’t really care what the formula for AVG is, or what the value of AVG exactly is, just that it IS (Sounds like Bill Clinton doesn’t it?).
We also know, and saw demonstrated above, that if we simply multiply AVG by the number of terms N (call it CNT) we get the value of the series out to PN (choose any N).
And we also know, and discussed above, that the number of terms (CNT) is obtained so very simply via the PNT (Prime Number Theorem) like so…
CNT = ______
Remember that I said earlier that I expected the Prime Number Theorem (PNT) to play a key role; it just happened! Anyway…
SeriesValue = AVG x CNT = AVG x __________
As N goes to ∞ (and thus PN goes to infinity) so does the Series Value (the sum of the terms) thus proving the divergence of the Harmonic Series of Primes.
So that’s the story of the search for a SIMPLE proof of the divergence of the Harmonic Series of Primes. Whew!!
Just The Proof
So considering all that background and explanation above,
here is the proof boiled down to just
5 Easy Pieces (the essentials):
But first we start with a reminder. The Harmonic Series of Primes (HSOP) is
out to 1/PN
1. There is an “Average Value” for the terms out to PN (we’ll call it AVG) which is the sum of the terms divided by the number of terms.
2. The value of the series out to PN = AVG x (The number of terms out to PN)
3. From the Prime Number Theorem, the number of terms (call it CNT) out to PN is
4. Thus, The value of the series (the sum of the terms) out to PN is
SeriesValue = AVG x CNT = AVG x PN
5. As N (and thus PN) goes to infinity so does CNT ( which per the PNT is PN / ln(PN) ) which gives us
SeriesValue = AVG x CNT = AVG x infinity
Which, of course, means that the Harmonic Series of Primes diverges.
Open The Gates!