In arithmetic series, like 1 + 2 + 3 + 4 + 5 + … , each term differs from the previous one by a fixed amount. There is a formula for calculating the sum of the first N terms. For geometric series, like 3 + 6 + 12 + 24 + … , each term is a fixed multiple of the previous one. Again, there is a formula for the sum of the first N terms of such a series.
Another series arises when we take the inverse of each term in an arithmetic series. For the simplest arithmetic series, 1 + 2 + 3 + 4 + 5 + … , the corresponding harmonic series is
1 + ½ + ⅓ + ¼ + ⅕ + … .
This is more complicated. There is no general formula for the sum of N terms. We have to calculate it by hand or use a calculator or computer. As is well known, the series diverges, but very slowly.
The shaded area between 1 and n gives the sum of the harmonic series to term (1/n).
Breaking Records
Harmonic series turn up in many contexts. We will describe their application to record breaking. Let us look at the mean annual temperature in Ireland. For now we will forget about climate change and assume that the weather varies randomly but without any systematic drift or trend. How often, on average, would we expect to have a record high temperature.
If we have temperature data for only one year, it is automatically “the hottest year on record” (as well as the coldest). If we now get data for the following year, there is a fifty-fifty chance that it is warmer than the first. The number of record years is now either 1 or 2. Taking the average, since we want a probability, we have 1 + ½ .
Now a third year’s data comes in and we calculate the mean temperature. There is a one-in-three chance that it is the hottest of the three, so the expected number of record-breaking years is now 1 + ½ + ⅓ . You can see where this is going: the expected number of record-breaking years in a series of mean temperatures for n years is 1 + ½ + ⅓ + ¼ + ⅕ + … + 1/n, the sum of the first n terms of the harmonic series. This number is called the n-th harmonic number. Let us denote it by
H(n) = 1 + ½ + ⅓ + ¼ + ⅕ + … + 1/n .
How does H(n) behave as n gets bigger? If you calculate H(n), you find that it grows ever more slowly with n. In fact, the sum diverges, but it creeps very gradually towards infinity. The first term is 1. The sum of the first 10 is 2.9. With 100 terms we reach 5.2 and with 1000 only 7.5.
Random Numbers
We took one hundred sequences, each with 1024 random numbers, and counted the number of records in each. The figure below shows the number of cases in which each number of records occurs. We note that H(1024) = 7.51 so we expect seven or eight record-breaking values on average. Indeed, the clear maximum number on the graph is for 8 record-breaking values.
Number of record values for 100 sequences, each with 1024 values. H(1024)=7.51.
Temperature Records in a Warming World
In ten years we would expect the temperature record to be broken about 3 times, since H(10)=2.9. In a century there would be about 5 new records, since H(100)=5.2. It is clear that records are more frequent in the early stages of the temperature record. However, in a steady-state climate, no matter how extreme the current record is, it is certain to be broken eventually, although the waiting time may be very large.
Of course, the real climate is changing and the mean temperature is rising inexorably. We need more than the simple harmonic series to deal with that!
To be continued …
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ThatsMaths 