Understanding the concept of convergence is crucial in conventional mathematics. It dictates that the distance of a series' partial sums to its hypothesized limit must decrease monotonically, and this distance can be made arbitrarily small. This principle underpins many mathematical models and is a fundamental tool in understanding the world around us.

The beauty of mathematics is that it can be used to create models that accurately describe real-world phenomena. There are other ways of modeling phenomena -- graphic, verbal, etc., but these models don't consistently respond in the same ways as mathematical models do.

Unfortunately, I don't know enough about string theory to comment on whether this series's claim to converge to -1/12 is useful in that area. My comment applies only to conventional math.

However, it is possible to select a set of self-consistent axioms and develop a model that contradicts what we generally perceive. Lewis Carroll, the pen name of mathematician Charles Lutwidge Dodgson, satirized this in his Alice stories.