It's time for the harmonic series.
[MUSIC].
That series has a name. This infinite series.
The sum of the reciprocals of the
positive whole numbers, is called the harmonic series.
I want to know if that series diverges or converges.
And the first thing to do is to look at the limit of the terms.
The nth term of the harmonic series is 1 over n.
So, if I take the limit as
n goes to infinity of the nth term, what's the
limit of 1 over n as n goes to infinity?
This is zero.
And what did the limit test say?
The limit tells us that if the limit of the nth term
of some series is not equal to zero, then the series diverges.
But, when the limit is equal to zero, that.
So in this case, the limit of the terms is zero.
And the limit test is silent.
At this point, we still don't know whether the harmonic series converges or diverges.
Well, I can start adding up a bunch of terms.
one plus a half.
The first two terms in the harmonic series is, three halves.
If I add the next term to the harmonic series.
One plus a half plus a third. That's 11/6th.
And I could add the next term on the harmonic series.
1 plus a half plus a third plus a fourth.
Well, that's 25/12.
I could add the next term in the harmonic series.
One plus a half plus a third plus a fourth, plus a fifth.
That's 137/60. And I could keep going like this.
And if I add the first ten terms together.
1/2, third, fourth, and so on, up to 1/10. I get a number that's just
under three. Let's add up even more terms.
So instead of just ten terms, I'll add up the first 100 terms.
And I'll get a number, you know, 5.187 or so.
The question is what happens when I add up more and more terms in this series?
Even more terms, well if I add up the first
1,000 terms, I get a number that is about 7.485.
If I add up the first 10,000
terms, I get a number that is just under ten.
We're adding up a ton of terms.
And still, the partial sums just aren't that big.
Let's take a different approach.
Here, I've started writing out the harmonic series, right?
One plus a half plus a third.
I've written down the first 32 terms of the harmonic series.
And of course, it keeps going.
I'm going to group the terms in a clever way.
Well, I'll set aside the one and
this first one half.
And then I'll put these next two terms together, the third and the fourth.
And I'll put these next four terms together.
A fifth plus a sixth plus a seventh plus an eighth.
And I'll put these next eight terms together.
The terms between a ninth and a sixteenth.
And I'll put these next 16 terms together between a 17th and a 32nd, and so on.
Now, let's underestimate
each of these groups.
I can underestimate 1/3 by replacing 1/3 with something smaller than 1/3, like 1/4.
1/4 is smaller than 1/3.
I can underestimate a fifth, a sixth and a seventh by replacing each
of those with something smaller than a fifth, a sixth, and a seventh.
Well, and eighth is smaller than a fifth. And eighth is smaller than a sixth.
And an eighth is smaller than a seventh.
Why is this a good idea?
Well, here I got two fourths, and two fourths is a half.
And that means a third plus a fourth if I add those together is at least a half.
Here I got four eighths.
Four eights is also a half, and that means a fifth, plus a sixth, plus a seventh,
plus and eighth, which is bigger than an eighth
plus an eighth, plus an eight, plus an eighth.
A fifth plus a sixth plus a seventh plus and eighth must be bigger than 4/8.
It must be bigger than one half.
I've got eight more terms here, starting at a ninth and ending at a sixteenth.
Each of these terms can be underestimated by a sixteenth, right?
A ninth is, bigger than a sixteenth. A tenth is bigger than a sixteenth.
An 11th is bigger than a 16th, a 12th is bigger than a
16th, all these terms are bigger than a 16th.
But now I've got eight 16th, and that means a 9th plus a 10th
plus a 16th, all of these together must be at least eight 16th.
And eight 16th.
Is one half.
So, if I add up these eight terms in the harmonic series, I get at least a half.
Now the next sixteen
terms in the harmonic series are each at least a thirty-second.
But if I add up sixteen terms each of which is at
least a thirty-second, that's an answer which is at least one half.
The next 32 terms in the harmonic series
starting at one 33rd and ending at one 64th.
Those 32 terms are at least a 64th. But 32 64th is a half, and that
means the sum of the next 32 terms in the harmonic series is at least a half.
The next 64 terms of the harmonic series starting
at the sixty-fifth and ending at the one over 128th.
Those 64 terms added up is at least 64 128th.
It's at least one half.
So when I'm adding up the harmonic series I
can underestimate the answer by adding one plus a half.
Plus a half, plus a half, plus a half, plus a half, and so on.
The summing the harmonic series is even worse than adding up
a half, and a half, and a half, and a half.
All that is to say, the harmonic series diverges.
Let's summarize this.
The harmonic series diverges Even though the size of the terms
gets very small, even though the limit of the nth term
is zero, and this isn't a contradiction.
The fact that we know that if a series
converges, then the limit of the nth term is zero.
But just because the limit of the nth term is zero doesn't mean the series converges.
The harmonic series is a great example of
this phenomenon, where limit of the nth
term is zero, but the series never the less, diverges.