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?