This morning, my friend Dr24Hours pinged me on twitter about some bad math:
— Dr24hours (@Dr24hours) January 17, 2014
And indeed, he was right. Phil Plait the Bad Astronomer, of all people, got taken in by a bit of mathematical stupidity, which he credulously swallowed and chose to stupidly expand on.
Let’s start with the argument from his video.
We’ll consider three infinite series:
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ... S2 = 1 - 2 + 3 - 4 + 5 - 6 + ... S3 = 1 + 2 + 3 + 4 + 5 + 6 + ...
S1 is something called Grandi’s series. According to the video, taken to infinity, Grandi’s series alternates between 0 and 1. So to get a value for the full series, you can just take the average – so we’ll say that S1 = 1/2. (Note, I’m not explaining the errors here – just repeating their argument.)
Now, consider S2. We’re going to add S2 to itself. When we write it, we’ll do a bit of offset:
1 - 2 + 3 - 4 + 5 - 6 + ... 1 - 2 + 3 - 4 + 5 + ... ============================== 1 - 1 + 1 - 1 + 1 - 1 + ...
So 2S2 = S1; therefore S2 = S1=2 = 1/4.
Now, let’s look at what happens if we take the S3, and subtract S2 from it:
1 + 2 + 3 + 4 + 5 + 6 + ... - [1 - 2 + 3 - 4 + 5 - 6 + ...] ================================ 0 + 4 + 0 + 8 + 0 + 12 + ... == 4(1 + 2 + 3 + ...)
So, S3 – S2 = 4S3, and therefore 3S3 = -S2, and S3=-1/12.
So what’s wrong here?
To begin with, S1 does not equal 1/2. S1 is a non-converging series. It doesn’t converge to 1/2; it doesn’t converge to anything. This isn’t up for debate: it doesn’t converge!
In the 19th century, a mathematician named Ernesto Cesaro came up with a way of assigning a value to this series. The assigned value is called the Cesaro summation or Cesaro sum of the series. The sum is defined as follows:
Let . In this series, . is called the kth partial sum of A.
The series is Cesaro summable if the average of its partial sums converges towards a value .
So – if you take the first 2 values of , and average them; and then the first three and average them, and the first 4 and average them, and so on – and that series converges towards a specific value, then the series is Cesaro summable.
Look at Grandi’s series. It produces the partial sum averages of 1, 1/2, 2/3, 2/4, 3/5, 3/6, 4/7, 4/8, 5/9, 5/10, … That series clearly converges towards 1/2. So Grandi’s series is Cesaro summable, and its Cesaro sum value is 1/2.
The important thing to note here is that we are not saying that the Cesaro sum is equal to the series. We’re saying that there’s a way of assigning a measure to the series.
And there is the first huge, gaping, glaring problem with the video. They assert that the Cesaro sum of a series is equal to the series, which isn’t true.
From there, they go on to start playing with the infinite series in sloppy algebraic ways, and using the Cesaro summation value in their infinite series algebra. This is, similarly, not a valid thing to do.
Just pull out that definition of the Cesaro summation from before, and look at the series of natural numbers. The partial sums for the natural numbers are 1, 3, 6, 10, 15, 21, … Their averages are 1, 4/2, 10/3, 20/4, 35/5, 56/6, = 1, 2, 3 1/3, 5, 7, 9 1/3, … That’s not a converging series, which means that the series of natural numbers does not have a Cesaro sum.
What does that mean? It means that if we substitute the Cesaro sum for a series using equality, we get inconsistent results: we get one line of reasoning in which a the series of natural numbers has a Cesaro sum; a second line of reasoning in which the series of natural numbers does not have a Cesaro sum. If we assert that the Cesaro sum of a series is equal to the series, we’ve destroyed the consistency of our mathematical system.
Inconsistency is death in mathematics: any time you allow inconsistencies in a mathematical system, you get garbage: any statement becomes mathematically provable. Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.
What makes this worse is that it’s obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn’t matter that infinity is involved: you can’t following a monotonically increasing trend, and wind up with something smaller than your starting point.
Someone as allegedly intelligent and educated as Phil Plait should know that.