Friday, 27 July 2012

World records and convergent sequences

Will world record times continue tumbling for ever without limit?

An infinite sequence of numbers can either converge or diverge.

An example of a divergent sequence would be: 1, 1.1, 1.2, 1.3, 1.4, 1.5 ... and so on, with the rule being 'add 0.1' each time to get the next number. Eventually the numbers in this sequence will get greater than any number we might write down.

An example of a convergent series would be: 1, 1.1, 1.11, 1.111, 1.1111, 1.11111 ... and so on. The rule here is easier to see visually than to express mathematically. Informally, what we do is to add another digit 1 after the decimal point. If you want a formula, the rule for the third number on would be: add one tenth of the difference between the last two numbers. So, to get the fourth number, for example, you add to 1.11 a tenth of (1.11 – 1.1), which is 'add 0.001'. This is a convergent sequence because the numbers get closer and closer to 1¹/₉. In fact they will eventually get as close to 1¹/₉ as you like.

I mention all this because we are about to get into the Olympic Games, and records will be broken again. This always raises the question as to whether there is a limit to how fast an athlete can run, say, the 100 metres? Will the records just go on and on for ever getting lower and lower without limit?

Well, mathematically, I can say with confidence that the sequence generated by the world record time year by year is definitely a convergent sequence. Since 2012, the world record times in seconds (to two decimal places) for the 100 metres on the 1st January each year generate this sequence:

9.79, 9.79, 9.79, 9.78, 9.78, 9.78, 9.77, 9.77, 9.74, 9.69, 9.58, 9.58, 9.58 ...

These times are given to two decimal places, that is, they are measured to the nearest 100th of a second. The current standard is to measure to the nearest thousandth of a second and then to round this to the nearest hundredth.

The numbers in this sequence by definition can never increase. A new record can only be lower than the previous record! So it is a non-increasing sequence. It also has a lower bound. We don't know how low the numbers in this sequence will get, but we do know that it is absolutely impossible for the numbers to get lower than zero! A theorem in the mathematics of infinite sequences is that a non-increasing sequence with a lower bound must be convergent.

This means that the world record time for the 100 metres must be converging on a lower limit. We don't know what it is and we will never know when we have reached it. But there is a limit. Be reassured: this is a convergent sequence.

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