Infinitely positive is neither infinite nor positive?

There's a recently publicized British biographical drama called "The Man Who Knew Infinity" about Srinivasa Ramanujan (1887-1920), a mathematician who, after growing up poor in Madras, India, earns admittance to Cambridge University and becomes a pioneer in mathematics. One of the things Ramanujan is particularly remembered for is adding an infinite number of (positive) things together, but getting something that is neither infinite, nor positive. For example: 1 + 2 + 3 + 4 + ... = -1/12

Every time this comes up, somebody asks me how adding an infinite number of progressively larger things can come out negative. Shouldn't the right answer be infinity?

It turns out (depending on what the dots mean) that both answers are correct. To understand how this is possible, suppose Achilles is out for a run and after spotting a wandering tortoise 100 meters ahead, he decides to catch up with it, only he plays a game of catching up to where the tortoise is now rather than where it will be. By the time Achilles catches up to where the tortoise was, the tortoise is 50 meters away. If Achilles again catches up to where the tortoise was, that sneaky tortoise is now 25 meters away. Achilles can keep playing this game indefinitely; to actually catch the tortoise would require catching up to where the tortoise was an infinite number of times.

Despite the infinite number of times Achilles catches up to the previous location of the tortoise, Achilles will, in fact, catch it. The trick is that these distances (as well as the times it takes for Achilles to cover them), get smaller and smaller. If Achilles runs at 1m/s and the tortoise at ½m/s, the infinite sum works out so the duo will meet after 100 seconds and Achilles has run 200 meters. (100m + 50m + 25m + ... = 200m)

This scenario, sometimes referred to as Zeno's Paradox, embodies what mathematicians call a "convergent series." The word "series" means all the terms are being added together and convergent means the sum approaches some number as terms are added. Usually, when terms get smaller and smaller, the series converges (though there are a few exceptions such as the harmonic series.) Convergent series are why, for example, a ball could theoretically bounce an infinite number of times before coming to rest a few seconds later.

Ramanujan's infinite sum is what mathematicians call a "divergent series" because the terms get bigger and bigger so the sum just blows up to infinity. Divergent series can apply to Achilles and the tortoise as well. Suppose Achilles is 100 meters ahead of the tortoise. How long until he catches the tortoise?

• A very good answer is "never." If the race were infinitely long, there's just no way for Achilles to catch the tortoise because Achilles already is ahead of it.
• Another very good answer is "in the past" since Achilles passed the tortoise 100 meters before the starting line 100 seconds before the race started. An equivalent way of saying this is that Achilles will catch the tortoise after negative one-hundred seconds (-100s) having traveled a distance of negative one-hundred meters (-100m).

So what does this have to do with Ramanujan and his infinite sum? Well, the tortoise can catch up to where Achilles was, and he can do this again and again... forever. It's just like before where Achilles closed in on the tortoise, only now the distances (as well as the time it takes to cover them) get bigger and bigger. Adding all these distances together makes a divergent series. Evaluating this sum, it makes sense to get an answer that's infinite... but it also makes sense to get an answer that's both non-infinite and negative. Similar logic can be applied to Ramanujan's sum.

Showing how Ramanujan's sum comes out to -1/12 takes a few more tricks. See below for a ad-hoc derivation: