5.3 Decreasing sequences

In order to examine the infinitely small – whether instants in time or precise positions in space – Newton and Leibniz looked at sequences of numbers. Some sequences get closer and closer to one number. Some do not, and instead they grow forever. Newton and Leibniz looked at these sequences of numbers and asked: what are they going towards? If you know that, you can develop a formal and robust way of talking about what happens in the instant, or in the infinitely small point. You do this by talking about how sequences of numbers that approach this value behave. Look again at the sequence of numbers you used earlier:

The fractions in this sequence are getting smaller and smaller. The sequence is going towards zero – but if you stop the sequence at any finite point, it hasn’t yet reached zero. Not all sequences that get smaller and smaller go towards zero. So how do you know that this one does? Newton and Leibniz gave us a formal way of saying what a sequence goes towards. This is called the ‘limit’ of a sequence.

The limit of the sequence above is zero. The way this is proved is by showing that, no matter how small a gap you look for, there’s a number in the sequence which is closer to zero. All subsequent numbers in the sequence are closer than that gap to zero.

In order for an infinite list of numbers to give us a finite answer when added together, you first need to know that the sequence goes towards zero. If not, then the sum is going to be unbounded and infinite.

There’s a second condition that’s necessary for an infinite list of numbers to give us a finite answer when added together. This condition is that the infinite list needs to go towards zero quickly enough. There are infinite lists that go towards zero, but nevertheless cannot be added together to give a finite sum. One example is:

Although the numbers go towards zero, they do not do so quickly enough, and you can show that their sum is infinite.

Using calculus, you can look at the infinitely small by looking at the behaviour of infinite sequences that get smaller and smaller. Calculus is used to look at any process that happens over time. It’s used in so many different fields: all the physical sciences, computer science, statistics, economics, engineering, medicine, and more. You can look at change over time, change in the instant of time, and you can look at the accumulation of what has happened over a period of time.

Typically, people meet calculus for the first time when asked to look at a rate of change for a phenomenon that has a changing rate of change: speed, rates of infection, value of share prices, etc. In the background there is the mathematics of infinite sequences that get smaller and smaller, allowing us to look at ever smaller intervals of time, and to know what’s happening at the infinitely small instant of time. It’s quite remarkable that the mathematics of the infinite provides practical solutions for everyday life in this way, as well as solutions to age-old paradoxes.