New world records in some sports are remarkably elusive, yet in others they seem to fall regularly, as Armand Duplantis's extraordinary pole vault recently demonstrated. Looking to records in other areas such as climate change can help explain why.

To the roar of the crowd at the Athletics World Championships in Tokyo, Armand "Mondo" Duplantis sprinted down the runway with his pole aloft. Pumping music reached a crescendo and the cheers became deafening. Then Duplantis flew.

The Swedish pole vaulter soared above the competition to assert his dominance over one of athletics' most technical disciplines – achieving a new world record in the event. His 6.3m (20.6ft) leap saw Duplantis win his third consecutive world title and, astonishingly, break his 14th world record. He is, in every way, raising the bar again and again. (Read more about how Armand Duplantis is reaching new heights from BBC Sport.)

This is often what can happen in sport. Improvements in diet, technique or equipment can bring a sudden spate of world records, especially in technical disciplines like pole‑vaulting and cycling where small adjustments make a big difference. Technical advances in running events have also seen records tumble in recent years.

In contrast, the men's world record in the long jump has been broken only once since 1968 when Mike Powell leapt 8.95m (29ft 4in) at the 1991 World Championships, also held in Tokyo. Perhaps we have reached peak long jump – a situation in which further improvement is impossible and differences between an athlete's performances come down to "luck": the wind, how well they slept the night before, and so on.

We refer to these sorts of situations as "stationary", in the sense that the overall trend in average behaviour is unchanging. Given a stationary system, we can ask how often we should expect records to fall due to random fluctuations. For an example of a stationary system on which we have long‑standing measurements, we might look to pre‑industrial climate data.

Imagine we investigate annual rainfall in different cities across the world, with the totals in each independent of the others and with no overall trend in climate behaviour. The first year's rainfall in each city is, by definition, a record. If the second year’s rainfall is independent of the first, then on average, half the cities will see rainfall exceed the first year while in half it will not. So after two years the average number of records across the cities is one plus a half.

In the third year, a total must exceed both previous years to set a new record. On average, this happens in only one-third of the cities, so by the end of year three the expected number of records is one plus a half plus a third. And so the series continues. After 100 years the expected number of records per city is one plus a half plus a third plus a quarter… plus a hundredth.

This set of consecutive fractions added together is known in mathematics as the Harmonic series because of........

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