WEBVTT
00:00:01.740 --> 00:00:04.559
Hi, yesterday we looked at two infinite series.
00:00:04.799 --> 00:00:08.339
Both had destinations, but one approached its
00:00:08.339 --> 00:00:11.859
destination painfully slowly, while the other
00:00:11.859 --> 00:00:15.720
got there remarkably fast. So we fine -tuned
00:00:15.720 --> 00:00:18.500
our question. It was not enough to ask, does
00:00:18.500 --> 00:00:22.059
this infinite series converge? We also asked,
00:00:22.179 --> 00:00:26.019
how quickly does it converge? So today, let's
00:00:26.019 --> 00:00:28.940
fine -tune things one step further. Let's consider
00:00:28.940 --> 00:00:33.479
this infinite sum. We have 1 over 1 plus 1 over
00:00:33.479 --> 00:00:38.320
2 plus 1 over 3 plus 1 over 4 plus dot dot dot.
00:00:38.960 --> 00:00:42.479
We call this the harmonic series. And notice
00:00:42.479 --> 00:00:45.259
what's happening. Each term gets smaller and
00:00:45.259 --> 00:00:49.060
smaller. Eventually, the individual terms gets
00:00:49.060 --> 00:00:53.960
as close to zero as we want. So today's question
00:00:53.960 --> 00:00:57.560
is if the individual pieces of an infinite sum
00:00:57.560 --> 00:01:01.649
keep shrinking towards zero, Is that enough to
00:01:01.649 --> 00:01:06.030
make the entire sum settle towards a finite number?
00:01:06.890 --> 00:01:11.010
Now, it certainly feels like it should. But the
00:01:11.010 --> 00:01:15.209
answer to this question is no. The harmonic series
00:01:15.209 --> 00:01:22.349
grows forever. Now, granted, very slowly, but
00:01:22.349 --> 00:01:26.530
without bound. Okay, now let's make one tiny
00:01:26.530 --> 00:01:30.000
adjustment. Rather than using the counting numbers
00:01:30.000 --> 00:01:33.299
in the denominator, let's square them. So we
00:01:33.299 --> 00:01:38.140
have 1 over 1 squared plus 1 over 2 squared plus
00:01:38.140 --> 00:01:42.459
1 over 3 squared plus 1 over 4 squared plus dot
00:01:42.459 --> 00:01:48.000
dot dot. Now the series converges. On a side
00:01:48.000 --> 00:01:52.780
note, it converges to pi squared over 6. And
00:01:52.780 --> 00:01:56.680
this is the famous Basel problem that Euler solved.
00:01:57.420 --> 00:02:00.640
And on another side note, I fussed months over
00:02:00.640 --> 00:02:04.120
creating a video on the Basel problem that you
00:02:04.120 --> 00:02:07.359
can watch at intersectingus .com if you're interested.
00:02:08.479 --> 00:02:12.620
But returning to our problem today, we can fine
00:02:12.620 --> 00:02:15.639
-tune this even more. What if we don't go all
00:02:15.639 --> 00:02:18.719
the way from an exponent of 1 to an exponent
00:02:18.719 --> 00:02:23.699
of 2? What if we made the exponent 1 .01? Then
00:02:23.699 --> 00:02:27.300
we have 1 divided by 1 raised to 1 .01 power
00:02:27.300 --> 00:02:32.240
plus 1 divided by 2 raised to 1 .01 power plus
00:02:32.240 --> 00:02:36.259
1 divided by 3 raised to 1 .01 power plus dot
00:02:36.259 --> 00:02:42.219
dot dot. What does this series do? Well, it converges
00:02:42.219 --> 00:02:46.060
as well. In fact, for this family of series,
00:02:46.280 --> 00:02:51.860
any exponent that exceeds 1 gives us convergence.
00:02:52.620 --> 00:02:56.819
So an exponent of 1 or less gives us divergence.
00:02:57.400 --> 00:03:01.460
That means exponent 1 diverges, but exponent
00:03:01.460 --> 00:03:08.520
1 .0000001 converges. And you can make that difference
00:03:08.520 --> 00:03:12.159
as small as you want. As long as the exponent
00:03:12.159 --> 00:03:18.500
exceeds 1, we have crossed the boundary. Isn't
00:03:18.500 --> 00:03:22.240
that interesting? I think it's remarkable. On
00:03:22.240 --> 00:03:24.819
both sides of the boundary, every individual
00:03:24.819 --> 00:03:28.439
term approaches zero. And to our eyes, the two
00:03:28.439 --> 00:03:32.280
series can look almost identical. And yet their
00:03:32.280 --> 00:03:38.719
ultimate behavior is completely different. Maybe
00:03:38.719 --> 00:03:41.759
that is fine -tuning in its purest mathematical
00:03:41.759 --> 00:03:46.159
form. Sometimes precision matters because the
00:03:46.159 --> 00:03:49.259
boundary between two completely different outcomes
00:03:49.879 --> 00:03:54.919
isn't wide at all. It is a line. On one side,
00:03:54.979 --> 00:03:58.599
the sun grows forever. On the other, it settles
00:03:58.599 --> 00:04:02.639
towards a finite destination. And no matter how
00:04:02.639 --> 00:04:08.840
closely we zoom in, that boundary remains. So
00:04:08.840 --> 00:04:10.879
tomorrow, we'll find another place where our
00:04:10.879 --> 00:04:15.319
everyday intuition needs some fine -tuning. We'll
00:04:15.319 --> 00:04:19.000
ask a seemingly simple question. What does it
00:04:19.000 --> 00:04:22.899
mean for an infinite set to be big?
00:00:01.740 --> 00:00:04.559
Hi, yesterday we looked at two infinite series.
00:00:04.799 --> 00:00:08.339
Both had destinations, but one approached its
00:00:08.339 --> 00:00:11.859
destination painfully slowly, while the other
00:00:11.859 --> 00:00:15.720
got there remarkably fast. So we fine -tuned
00:00:15.720 --> 00:00:18.500
our question. It was not enough to ask, does
00:00:18.500 --> 00:00:22.059
this infinite series converge? We also asked,
00:00:22.179 --> 00:00:26.019
how quickly does it converge? So today, let's
00:00:26.019 --> 00:00:28.940
fine -tune things one step further. Let's consider
00:00:28.940 --> 00:00:33.479
this infinite sum. We have 1 over 1 plus 1 over
00:00:33.479 --> 00:00:38.320
2 plus 1 over 3 plus 1 over 4 plus dot dot dot.
00:00:38.960 --> 00:00:42.479
We call this the harmonic series. And notice
00:00:42.479 --> 00:00:45.259
what's happening. Each term gets smaller and
00:00:45.259 --> 00:00:49.060
smaller. Eventually, the individual terms gets
00:00:49.060 --> 00:00:53.960
as close to zero as we want. So today's question
00:00:53.960 --> 00:00:57.560
is if the individual pieces of an infinite sum
00:00:57.560 --> 00:01:01.649
keep shrinking towards zero, Is that enough to
00:01:01.649 --> 00:01:06.030
make the entire sum settle towards a finite number?
00:01:06.890 --> 00:01:11.010
Now, it certainly feels like it should. But the
00:01:11.010 --> 00:01:15.209
answer to this question is no. The harmonic series
00:01:15.209 --> 00:01:22.349
grows forever. Now, granted, very slowly, but
00:01:22.349 --> 00:01:26.530
without bound. Okay, now let's make one tiny
00:01:26.530 --> 00:01:30.000
adjustment. Rather than using the counting numbers
00:01:30.000 --> 00:01:33.299
in the denominator, let's square them. So we
00:01:33.299 --> 00:01:38.140
have 1 over 1 squared plus 1 over 2 squared plus
00:01:38.140 --> 00:01:42.459
1 over 3 squared plus 1 over 4 squared plus dot
00:01:42.459 --> 00:01:48.000
dot dot. Now the series converges. On a side
00:01:48.000 --> 00:01:52.780
note, it converges to pi squared over 6. And
00:01:52.780 --> 00:01:56.680
this is the famous Basel problem that Euler solved.
00:01:57.420 --> 00:02:00.640
And on another side note, I fussed months over
00:02:00.640 --> 00:02:04.120
creating a video on the Basel problem that you
00:02:04.120 --> 00:02:07.359
can watch at intersectingus .com if you're interested.
00:02:08.479 --> 00:02:12.620
But returning to our problem today, we can fine
00:02:12.620 --> 00:02:15.639
-tune this even more. What if we don't go all
00:02:15.639 --> 00:02:18.719
the way from an exponent of 1 to an exponent
00:02:18.719 --> 00:02:23.699
of 2? What if we made the exponent 1 .01? Then
00:02:23.699 --> 00:02:27.300
we have 1 divided by 1 raised to 1 .01 power
00:02:27.300 --> 00:02:32.240
plus 1 divided by 2 raised to 1 .01 power plus
00:02:32.240 --> 00:02:36.259
1 divided by 3 raised to 1 .01 power plus dot
00:02:36.259 --> 00:02:42.219
dot dot. What does this series do? Well, it converges
00:02:42.219 --> 00:02:46.060
as well. In fact, for this family of series,
00:02:46.280 --> 00:02:51.860
any exponent that exceeds 1 gives us convergence.
00:02:52.620 --> 00:02:56.819
So an exponent of 1 or less gives us divergence.
00:02:57.400 --> 00:03:01.460
That means exponent 1 diverges, but exponent
00:03:01.460 --> 00:03:08.520
1 .0000001 converges. And you can make that difference
00:03:08.520 --> 00:03:12.159
as small as you want. As long as the exponent
00:03:12.159 --> 00:03:18.500
exceeds 1, we have crossed the boundary. Isn't
00:03:18.500 --> 00:03:22.240
that interesting? I think it's remarkable. On
00:03:22.240 --> 00:03:24.819
both sides of the boundary, every individual
00:03:24.819 --> 00:03:28.439
term approaches zero. And to our eyes, the two
00:03:28.439 --> 00:03:32.280
series can look almost identical. And yet their
00:03:32.280 --> 00:03:38.719
ultimate behavior is completely different. Maybe
00:03:38.719 --> 00:03:41.759
that is fine -tuning in its purest mathematical
00:03:41.759 --> 00:03:46.159
form. Sometimes precision matters because the
00:03:46.159 --> 00:03:49.259
boundary between two completely different outcomes
00:03:49.879 --> 00:03:54.919
isn't wide at all. It is a line. On one side,
00:03:54.979 --> 00:03:58.599
the sun grows forever. On the other, it settles
00:03:58.599 --> 00:04:02.639
towards a finite destination. And no matter how
00:04:02.639 --> 00:04:08.840
closely we zoom in, that boundary remains. So
00:04:08.840 --> 00:04:10.879
tomorrow, we'll find another place where our
00:04:10.879 --> 00:04:15.319
everyday intuition needs some fine -tuning. We'll
00:04:15.319 --> 00:04:19.000
ask a seemingly simple question. What does it
00:04:19.000 --> 00:04:22.899
mean for an infinite set to be big?
