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Unbelievable: 1+2+3+4…. = -1/12 ?

The above video by the excellent team at Numberphile has caused a bit of an internet stir – by providing a proof that 1+2+3+4+5+… = -1/12

It’s well worth watching as an example of what proof means – if something is proved which we “know” is wrong, then should we accept it as true?  The particular proof as offered in the video is certainly open to question – even if the end result: 1+2+3+4+5+… = -1/12 can actually be proved under certain definitions, using the Riemann Zeta function.

Grandi’s Series

The proof in the video requires that firstly we accept that the infinite summation, 1-1+1-1+1-1… = 1/2.  This series is known as the Grandi’s Series – and has been the cause of arguments in the mathematical community for centuries as to what the infinite summation should actually be.   One method (called Cesaro Summation) gives an answer of 1/2 – which is the answer accepted in the video.

Alternative interpretations of Grandi’s series would be to group the numbers as 1 + (-1+1) + (-1+1) +(-1+1)…. which you would expect to equal 1. Or, we could group the numbers as (1-1) + (1-1) + (1-1) … which you would expect to equal 0.  Therefore it would be also mathematically valid to say that the infinite summation 1-1+1-1…  has no sum.

Divergent Series are the invention of the Devil

For the proof in the video to be valid we have to therefore accept that the sum of Grandi’s series is 1/2.  We also need accept that it is possible to manipulate infinite series by “shifting them along by 1” or by factorising.

However as we have already seen in the case of Grandi’s series, infinite series don’t always follow normal arithmetic rules. Indeed, the 19th century Norwegian mathematician Niels Abel, warned that that, “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever!”

Nevertheless it is an interesting method.  First they define 3 different infinite series:

S =  1 + 2 + 3 + 4 + 5 …..

S₁ = 1 – 1 + 1 – 1 + 1 – 1 ….

S₂ = 1 – 2 + 3 – 4 + 5…

Step 1:

The first step is to state that S₁ = 1 – 1 + 1 – 1 + 1 – 1 …. = 1/2.

Step 2:

if S₂ = 1-2+3-4+5…
then 2S₂ = 1 -2+3-4+5…
+1-2+3-4…

Here we have “shifted along by one space” the second S₂. This means that when we add the two sequences together we end up with:

2S₂    = 1-1+1-1+1…  = 1/2
which gives S₂ = 1/4.

Step 3:

Do S – S₂ = 1 + 2 + 3 + 4 + 5 …..
-(1 – 2 + 3 – 4 + 5…)
= 4 + 8 + 12 + ….
= 4(1 + 2 + 3….)
= 4(S)

Now if S – S₂ = 4S we can simply rearrange this equation and substitute the value of S₂ = 1/4 which we found before to give: S = -1/12

As mentioned above this is not a very rigorous proof. There is a more rigorous (and complicated) method of proving this – which is the method used by Euler, and which employs the Riemann Zeta function. You can watch this method here:

You might notice when watching this proof that at the start of the video they use the infinite summation of a geometric sequence formula – which is only valid for absolute x less than 1. Then later on they substitute x = -1 into a result derived from it. This is OK because of analytical continuation (which is a method of extending the domain of a function beyond its usual domain). This idea starts to get really complicated – but if you’re interested in the basic idea look at the post on the Riemann Sphere below. The Riemann Sphere allows infinity to be included in the domain of the complex numbers.

If you enjoyed this post you might also like:

Mathematical Proof and Paradox: How you can “prove” things like 1 = 2. Can you spot the flaws in the logic?

The Riemann Hypothesis: How the Riemann Zeta function is fundamental to understanding the prime numbers – and how solving the Riemann Hypothesis is one of the greatest puzzles in mathematics.

The Riemann Sphere – an introduction to isomorphic mappings, which is a lot more interesting than it sounds!

Essential resources for IB students:

1) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.