Doing algebra with infinite series is very risky

[polymath257]

Well, the problem is, precisely, that the series is not convergent. So, it literally has no value. So the problem is in the very first step.

In those situations where it *is* convergent, the algebra works and the sum *is* -1. The 2-adics are an example of this.

So....make sure an expression has a value before doing algebra with it.