As promised I want to talk to-day about a little bit of Maths. As a Mathematician, I think that there are some things which more people should know – beyond “Fermat’s Last Theorem” or “Pythagoras'” from what they may have learned at school.
Today I am going to talk a bit about sequences. Here in the UK, most students will learn about arithmetic progressions (things like 1, 4, 7, 10 where the difference between terms is constant) and geometric progressions (things like 1, 3, 9, 27 where the ratio between terms is constant) in any A-Level course. These have related series (where we add up terms of the sequences). And hopefully most students will have learned the formulas for each. One area where pupils seem to get confused is when we start adding up an infinite number of terms of a sequence – infinite series are very weird things indeed!
There are also other types of sequences, such as alternating sequences (where the terms are, as the title suggests, alternating their sign) and let us look at the following alternating series.
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 – … (where the … mean continuing on for ever)
Well what does this look like? What is the sum if we assume there are infinitely many terms?
We can consider the partial sums – that is, the finite sums, which becomes 1, 0, 1, 0, 1, 0, 1, 0 etc. so if would appear that the answer has to be 1 or 0 depending on how many terms we are adding. But infinity is not a number – it is a concept (again there will probably be a similar blog post on this confusion in the coming months!).
Well, we can pair up the terms as follows
(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + … where each bracket is 0 and since there will be infinitely many zeros being added up the sum is 0 (or is it?)
Instead we could have paired up the terms as follows
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + … where each bracket is 0 and since there are infinitely many zeros being added then the sum is 1 (since we paired up from the second term rather than the first term)
So which is correct? Well, unsurprisingly, it depends on your definition; often in Mathematics it will depend on the context and here we have not been specific enough to say which is “correct” in the usual sense.
Let us assume that the sum exists, call it k. Well k = 1 – 1 + 1 – 1 + 1 – 1 + … and notice that we can then write
k = 1 – (1 – 1 + 1 – 1 + 1 – 1 + 1 – …) so k = 1 – k which is a simple equation 2k = 1 so k = 1/2
ARGH! So the sum, k, is 1/2. This certainly, on the face of it, seems more logical as we have only done a trick similar to that which shows 0.999… = 1 (see here) where we helped to eliminate the infinite string of 9s with a little mathematical magic. And in fact, this is the “accepted” sum. So if you add up infinitely many 1s and -1s then you obtain 1/2.
This series if called Grandi’s Series and has a rich history.
I hope you enjoyed this little introduction to some Mathematical magic which certainly seems to upset the usual way of thinking! Next time we will extend on some of these ideas and show something even more, apparantly, ludicrous (and explain why it makes sense) that
1 + 2 + 3 + 4 + 5 + … = -1/12
That will require Magic so I had better go shopping. Peace.