1 + 2 + 3 + 4 + ... = ?

I’ve already told you that I don’t really fully understand mathematics.  With that in mind, I came across a mathematical series known as the Ramanujan Summation the other day.  It’s named after the Indian mathematician Srinivasa Ramanujan and states that if you add all the natural numbers to infinity, 1 + 2 + 3 + 4 + 5 + 6…, you will find that it is equal to -1/12!  That seemed impossible to me, so I kept reading further.  The proof makes sense, even as someone who is not mathematically inclined. Please allow me to elucidate!

In order to build the proof for the Ramanujan Summation, I first need to convince you of the following mathematical proofs (trust me on this - you will see why at the end):

1 – 1 + 1 – 1 + 1 – 1 … = ½

1 – 2 + 3 – 4 + 5 – 6 … = ¼

Okay, are you ready? Let’s prove that the first infinite series above is equal to ½, which seemed fairly unlikely to me (at least until I read the proof).

Let A = 1 – 1 + 1 – 1 + 1 – 1 …

Now, let's use a trick from algebra and subtract both sides of the equation from 1 (we can do that if it's done on both sides of the equation):

1 – A = 1 – (1 – 1 + 1 – 1 + 1 – 1 …)

Now as my math teacher wife always tells her students, "When in doubt, simplify!"  Simplifying the above equation results in:
 
1 – A = 1 – 1 + 1 – 1 + 1 – 1 …
 
See that?  The series on the righthand side of the equation is actually our original series A.  Through a mathematical sleight of hand, we have shown that 1 – A = A.  Now, use simple algebra to simplify the rest:
 
1 – A = A
 
1 – A + A = A + A
 
1 = 2A

 
½ = A
 
In other words, we have shown that the sum of the original infinite series (which we called A) is ½, which doesn’t really make intuitive sense to me, but hey that’s mathematics!  Apparently this infinite series is named after the Italian mathematician Luigi Guido Grandi and is called Grandi’s series.
 
Now let’s tackle the second infinite series above.  We will follow the same method that we used to prove Grandi’s series:
 
Let B = 1 – 2 + 3 – 4 + 5 – 6 …
 
Let’s now use A to help us out.  We subtract both sides of the equation from A (remember that we can do this as long as we do it on both sides of the equation), as shown below:
 
A – B = (1 – 1 + 1 – 1 + 1 – 1 …) – (1 – 2 + 3 – 4 + 5 – 6 …)
 
A – B = (1 – 1 + 1 – 1 + 1 – 1 …) – 1 + 2 – 3 + 4 – 5 + 6…

Using more mathematical sleight of hand, we can rearrange the above equation to:
 
A – B = (1 – 1) + (-1 + 2) + (1 – 3) + (-1 + 4) + (1 – 5) + (-1 + 6)…
 
The above equation simplifies further to:
 
A – B = 0 + 1 – 2 + 3 – 4 + 5 …

A - B = 1 – 2 + 3 – 4 + 5 …
 
Note that the series on the right side of the equation above is actually our original series B.  So, substituting B where appropriate yields the following:
 
A – B = B
 
A – B + B = B + B

A = 2B

But wait!  We know that A = 1/2 (proved above).  Let's substitute 1/2 for A in the above equation, which yields:
 
½ = 2B

Now all we have to do is solve for B:
 
¼ = B
 
Tuh-dah!  There’s not a fancy name for this infinite series, but that’s okay.  It’s also apparently very well-known and commonly used throughout the fields of mathematics and physics.
 
Okay, let’s get back to the original point of this entire discussion, proving that the infinite series 1 + 2 + 3 + 4 + 5 … = -1/12.  Let’s start again by defining a new series:
 
C = 1 + 2 + 3 + 4 + 5… 
 
Let’s now subtract C from B (similar to what we’ve done in the previous two proofs) and simplify using more mathematical sleight of hand:
 
B – C = (1 – 2 + 3 – 4 + 5 – 6 …) – (1 + 2 + 3 + 4 + 5 + 6…)
 
B – C = (1 – 2 + 3 – 4 + 5 – 6 …) – 1 – 2 – 3 – 4 – 5 – 6…
 
B – C = (1 – 1) + (-2 – 2) + (3 – 3) + (-4 – 4) + (5 – 5) + (-6 + 6)…
 
B – C = 0 – 4 + 0 – 8 + 0 – 12 …
 
B – C = -4 – 8 – 12 …
 
Alright, that’s not what you were probably expecting, but it’s still a long way off from our result of -1/12.  Or is it?  Note that all the numbers on the righthand side of the equation are multiples of 4.  We can use simple algebra to factor the 4 out as follows:
 
B – C = -4 (1 + 2 + 3…)
 
Whoa!  Now we have our original series C on the righthand side of the equation.  We can now simplify further:
 
B – C = -4C
 
B = -3C

Again, we know that B = 1/4, so we can substitute and then solve for C:
 
¼ = -3C
 
-1/12 = C.
 
Voila! It seems impossible to me that the sum of all natural numbers up to infinity can equal not only a negative number but also a fraction!  However, the “proof is in the pudding” so to speak, and hopefully you can follow the intuition above.  Apparently the Ramanujan Summation is used throughout mathematics and physics, but that's for another day (okay, the likelihood that I will be writing anything more about the Ramanujan Summation is fairly low, so you will have to just research it yourself, if you are so inclined).  However, according to Wikipedia, a mathematician from the University of Alberta, Terry Gannon, calls the Ramanujan Summation "one of the most remarkable formulae in science" in a monograph on - wait for it - moonshine theory!  Who knew that mathematics and moonshine can be used in the same sentence!?!

Once again, things aren't always as obvious as they seem.  Just like the High Reliability Organization principle of Reluctance to Simplify!  The simple explanations are usually the wrong ones, particularly when it comes to complex systems.  The answer may be hidden beneath the surface.  Dig deeper.  Always take the next step.