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It's time for the harmonic series.

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[MUSIC].

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That series has a name.
This infinite series.

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The sum of the reciprocals of the

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positive whole numbers, is called the
harmonic series.

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I want to know if that series diverges or
converges.

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And the first thing to do is to look at
the limit of the terms.

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The nth term of the harmonic series is 1
over n.

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So, if I take the limit as

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n goes to infinity of the nth term, what's
the

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limit of 1 over n as n goes to infinity?

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This is zero.

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And what did the limit test say?

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The limit tells us that if the limit of
the nth term

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of some series is not equal to zero, then
the series diverges.

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But, when the limit is equal to zero,
that.

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So in this case, the limit of the terms is
zero.

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And the limit test is silent.

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At this point, we still don't know whether
the harmonic series converges or diverges.

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Well, I can start adding up a bunch of
terms.

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one plus a half.

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The first two terms in the harmonic series
is, three halves.

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If I add the next term to the harmonic
series.

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One plus a half plus a third.
That's 11/6th.

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And I could add the next term on the
harmonic series.

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1 plus a half plus a third plus a fourth.

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Well, that's 25/12.

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I could add the next term in the harmonic
series.

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One plus a half plus a third plus a
fourth, plus a fifth.

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That's 137/60.
And I could keep going like this.

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And if I add the first ten terms together.

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1/2, third, fourth, and so on, up to 1/10.
I get a number that's just

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under three.
Let's add up even more terms.

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So instead of just ten terms, I'll add up
the first 100 terms.

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And I'll get a number, you know, 5.187 or
so.

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The question is what happens when I add up
more and more terms in this series?

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Even more terms, well if I add up the
first

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1,000 terms, I get a number that is about
7.485.

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If I add up the first 10,000

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terms, I get a number that is just under
ten.

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We're adding up a ton of terms.

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And still, the partial sums just aren't
that big.

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Let's take a different approach.

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Here, I've started writing out the
harmonic series, right?

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One plus a half plus a third.

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I've written down the first 32 terms of
the harmonic series.

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And of course, it keeps going.

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I'm going to group the terms in a clever
way.

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Well, I'll set aside the one and

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this first one half.

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And then I'll put these next two terms
together, the third and the fourth.

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And I'll put these next four terms
together.

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A fifth plus a sixth plus a seventh plus
an eighth.

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And I'll put these next eight terms
together.

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The terms between a ninth and a sixteenth.

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And I'll put these next 16 terms together
between a 17th and a 32nd, and so on.

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Now, let's underestimate

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each of these groups.

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I can underestimate 1/3 by replacing 1/3
with something smaller than 1/3, like 1/4.

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1/4 is smaller than 1/3.

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I can underestimate a fifth, a sixth and a
seventh by replacing each

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of those with something smaller than a
fifth, a sixth, and a seventh.

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Well, and eighth is smaller than a fifth.
And eighth is smaller than a sixth.

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And an eighth is smaller than a seventh.

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Why is this a good idea?

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Well, here I got two fourths, and two
fourths is a half.

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And that means a third plus a fourth if I
add those together is at least a half.

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Here I got four eighths.

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Four eights is also a half, and that means
a fifth, plus a sixth, plus a seventh,

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plus and eighth, which is bigger than an
eighth

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plus an eighth, plus an eight, plus an
eighth.

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A fifth plus a sixth plus a seventh plus
and eighth must be bigger than 4/8.

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It must be bigger than one half.

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I've got eight more terms here, starting
at a ninth and ending at a sixteenth.

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Each of these terms can be underestimated
by a sixteenth, right?

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A ninth is, bigger than a sixteenth.
A tenth is bigger than a sixteenth.

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An 11th is bigger than a 16th, a 12th is
bigger than a

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16th, all these terms are bigger than a
16th.

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But now I've got eight 16th, and that
means a 9th plus a 10th

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plus a 16th, all of these together must be
at least eight 16th.

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And eight 16th.

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Is one half.

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So, if I add up these eight terms in the
harmonic series, I get at least a half.

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Now the next sixteen

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terms in the harmonic series are each at
least a thirty-second.

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But if I add up sixteen terms each of
which is at

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least a thirty-second, that's an answer
which is at least one half.

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The next 32 terms in the harmonic series

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starting at one 33rd and ending at one
64th.

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Those 32 terms are at least a 64th.
But 32 64th is a half, and that

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means the sum of the next 32 terms in the
harmonic series is at least a half.

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The next 64 terms of the harmonic series
starting

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at the sixty-fifth and ending at the one
over 128th.

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Those 64 terms added up is at least 64
128th.

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It's at least one half.

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So when I'm adding up the harmonic series
I

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can underestimate the answer by adding one
plus a half.

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Plus a half, plus a half, plus a half,
plus a half, and so on.

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The summing the harmonic series is even
worse than adding up

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a half, and a half, and a half, and a
half.

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All that is to say, the harmonic series
diverges.

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Let's summarize this.

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The harmonic series diverges Even though
the size of the terms

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gets very small, even though the limit of
the nth term

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is zero, and this isn't a contradiction.

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The fact that we know that if a series

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converges, then the limit of the nth term
is zero.

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But just because the limit of the nth term
is zero doesn't mean the series converges.

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The harmonic series is a great example of

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this phenomenon, where limit of the nth

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term is zero, but the series never the
less, diverges.

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[NOISE]