Good morning! It’s snowing today, it was sunny a few weeks ago, and who knows what will happen next. In this spirit of surprise, today we’ll look at 1+2+3+…. Any idea what it is if you keep adding? You might think you’d approach infinity, but actually….

….well, actually that makes sense. But then this would be mighty short, so instead we’ll prove that the sum is -1/12. This is called the Ramanujan Summation after the mathematician Srinivasa Ramanujan who was born in 1887 and who passed away in 1920.

So, let’s get proving! We’ll do this in parts:

**Step 1: ** Prove that 1-1+1-1+1-1+-… adds to 1/2.

We’ll call the sum of this sequence A, and do some fancy algebra:

Since A=1-1+1-1…. then if we subtract 1 (the first 1 on the right) we get:

A-1=-1+1-1+1…., which is the negative of what we started with. That means (A-1)=-A, so (2A-1)=0, and that means A=1/2. All done!

This all assumes that we can treat infinite sums the same way as finite sums. YMMV.

**Step 2:** Prove that 1-2+3-4+5-6+…. adds to 1/4.

We’ll call the sum of this sequence B, and keep going with the fancy algebra.

Since B=1-2+3-4+5-6+…. let’s look at A-B

A-B=(1-1+1-1+1-1+…)-(1-2+3-4+5-6+…)

Let’s reorder, putting the first terms together, the second terms, etc.

This is (1-1)+(-1+2)+(1-3)+(-1+4)+(1-5)+(-1+6)+…, which simplifies to

0+1-2+3-4+5. And that’s just B!

So A-B=B, which means A=2B, so B is half of A, and therefore 1/4.

**Step 3**: .Prove that 1+2+3+4+5+6+… adds to -1/12.

We’ll call this sequence C.

Since C=1+2+3+4+5+6+…, let’s look at B-C

B-C=(1-2+3-4+5-6+…)-(1+2+3+4+5+6+)

Like we did in Step 2, we’ll reorder, putting the first terms together, the second terms, etc.

This is (1-1)+(-2-2)+(3-3)+(-4-4)+(5-5)+(-6-6)+…, which simplifies to:

0-4+0-8+0-12+…., which is -4-8-12-…

You can factor out a -4, and get -4(1+2+3+…), and that’s -4C!

So B-C=-4C, giving B=-3C, so C is (-1/3) of B, or (-1/3) of (1/4) and that, my friends, is -1/12!

What do you think? If you think it makes sense, you’re in luck – there are some deep results in physics that use these ideas (although they are proved using something called the Riemann zeta function). On the other hand, if you think there was some mathematical sleight of hand, well, you’re right also. Treating infinite series like they are finite makes sense until it doesn’t, like adding up a bunch of positive integers and getting -1/12.

This subject was inspired by a reference in *The Art of Logic in an Illogical World* by Eugenia Cheng, and is also on a Numberphile video. I used a post on Cantor’s Paradise for the notation, and Scientific American for additional background.