WEBVTT 00:00:02.160 --> 00:00:05.599 Hello! Yesterday we saw an infinite process that 00:00:05.599 --> 00:00:09.060 approached something exact. 1 half plus 1 fourth 00:00:09.060 --> 00:00:12.300 plus 1 eighth plus 1 sixteenth plus dot dot dot. 00:00:12.939 --> 00:00:16.059 Even though it continued forever, the sum approached 00:00:16.059 --> 00:00:20.079 1 perfectly. Today, let's consider something 00:00:20.079 --> 00:00:23.460 that seems similar, but behaves completely different. 00:00:24.199 --> 00:00:27.339 Here is a sum that we refer to as the harmonic 00:00:27.339 --> 00:00:32.899 series. 1 over 1 plus 1 over 2. plus 1 over 3, 00:00:33.039 --> 00:00:37.759 plus 1 over 4, plus 1 over 5, plus dot dot dot. 00:00:38.880 --> 00:00:41.780 Just like yesterday, the terms keep getting smaller. 00:00:42.520 --> 00:00:45.399 And because of that, it is tempting to think 00:00:45.399 --> 00:00:48.840 this sum should also settle down to some finite 00:00:48.840 --> 00:00:53.399 number. But it doesn't. Even though the terms 00:00:53.399 --> 00:00:57.799 shrink, they do not shrink fast enough. And as 00:00:57.799 --> 00:01:01.259 we keep adding more and more terms, The total 00:01:01.259 --> 00:01:07.900 continues to grow. Slowly. Very slowly. But it 00:01:07.900 --> 00:01:11.959 never settles. It keeps increasing without bound. 00:01:12.400 --> 00:01:17.420 To infinity. The infinite sum from yesterday 00:01:17.420 --> 00:01:21.879 and today look almost the same. In both examples, 00:01:22.000 --> 00:01:25.379 we keep adding forever, the terms get smaller, 00:01:25.579 --> 00:01:29.579 and the process never ends. What I find interesting 00:01:29.579 --> 00:01:32.680 is one infinite sum approaches a finite number, 00:01:32.840 --> 00:01:37.420 1, and the other grows not only to 2 or 3 or 00:01:37.420 --> 00:01:42.519 even 100, but without bound. In mathematics, 00:01:42.739 --> 00:01:46.019 we say the first series converges and the second 00:01:46.019 --> 00:01:51.060 diverges. A small change in the pattern creates 00:01:51.060 --> 00:01:55.120 a completely different outcome. But the part 00:01:55.120 --> 00:01:58.230 I find most interesting is that this sum grows 00:01:58.230 --> 00:02:04.390 unbelievably slowly. To reach a total of 2 only 00:02:04.390 --> 00:02:10.229 requires 4 terms. To reach 3 requires 11 terms. 00:02:10.490 --> 00:02:17.050 To reach 4 takes 31 terms. Already, the slowdown 00:02:17.050 --> 00:02:21.430 is evident. But then it becomes dramatic. To 00:02:21.430 --> 00:02:27.710 reach 10, requires more than 12 ,000 terms. And 00:02:27.710 --> 00:02:33.810 to reach 100? Well, that requires an unimaginably 00:02:33.810 --> 00:02:39.229 enormous number of terms. At every stage, it 00:02:39.229 --> 00:02:43.490 feels like the sum is barely moving. You might 00:02:43.490 --> 00:02:46.770 think, surely, eventually, it must level off. 00:02:47.150 --> 00:02:50.659 But it never does. No matter how large a number 00:02:50.659 --> 00:02:53.280 you choose, this sum will eventually grow larger 00:02:53.280 --> 00:02:57.680 than that number. Now, my intuition is this seems 00:02:57.680 --> 00:03:02.340 impossible. But infinity has a different ending, 00:03:02.479 --> 00:03:07.960 which makes infinity feel even stranger. This 00:03:07.960 --> 00:03:10.379 is not merely something mathematicians suspect 00:03:10.379 --> 00:03:14.620 to be true. In mathematics, we can actually prove 00:03:14.620 --> 00:03:18.979 it with certainty. No matter what positive number 00:03:18.979 --> 00:03:23.520 we imagine, this infinite sum will eventually 00:03:23.520 --> 00:03:28.240 surpass it, even though the number of terms required 00:03:28.240 --> 00:03:35.439 may itself feel beyond imagination. Well, if 00:03:35.439 --> 00:03:38.000 you want to understand why the harmonic series 00:03:38.000 --> 00:03:41.680 continues without bound, you can read my article 00:03:41.680 --> 00:03:46.419 called A Log That Is Natural. It's on my Lazarus 00:03:46.419 --> 00:03:51.759 Math series at intersectingus .com. Okay, tomorrow 00:03:51.759 --> 00:03:54.340 we'll take a different perspective on infinity 00:03:54.340 --> 00:03:58.460 to see where it appears in surprising places.