Just because you don't understand doesn't mean it's "unsound pseudo-algebra." There are ways to properly manipulate infinite series, and Wikipedia does actually explain some of them on the page, even if it relies on a generally-unsound way to illustrate the intuition to a lay reader.

It is very well-known and long-recognized that our natural intuition is very wrong when it comes to infinities and infinite objects, and mathematics had a crisis in trying to come to grips with it.