You could say you just go round and round hunting for it, but no matter how hard you try you just can't corner it.

Well, you could.

there’s a way to tie it back to circles

Not necessarily circles, but conic sections. When you take a series of a fixed exponent over a variable x, and graph it, that graph is a parabola.

A parabola is a slice through a cone. Tada, pi appears.

But the first few values are:

1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28...

I really don't see any pattern there showing why it converges to 2 exactly

Edit:

After thinking some more, you could write the sum as:

(Sum from n=1 to infinity of): 2/(n * (n + 1))

That sum is smaller than the sum of:

2 * (1/n^2^) which converges to π^2^/3

So I can see why it converges, just not where to.

Not really. The left side comes from trying to evaluate the Reimann zeta function at a point outside its domain, and the right side comes from evaluating the analytic continuation of the zeta function at the same point. The deepest truth it reveals is that applying the definition of a function outside its domain can give you nonsense results.

It's one of those ideas like Scrödinger's cat or pulling yourself up by your bootstraps, where something that was intended to illustrate an absurdity is instead taken as representing some amazing and unintuitive truth.

Never understood those circumstances, but I choose to believe that all even numbers summed are -2/12 and all odd numbers therefore 1/12.

And 2 is.