Actually, his math was just fine for where math had developed at the time. Also, the guy's name was Luigi Guido Grandi, and he was an Italian monk, priest, mathematician, and engineer. The first link has a nice summary of both his life and this famous series.
The problem with this series is that it is not absolutely convergent. But these ideas were not well formulated, understood, of proven in his time. Even Leibnitz (co-inventor of the calculus) weighed in on the controversy of Grandi's series. It should also be noted that Grandi wasn't some hack but contributed to the spread of Leibnitz's calculus by introducing in Italy through his studies of a famous curve called (in English), "The Witch of Agnesi".
The particular "value" of the series doesn't exist, but it "should" be 1/2. These two conclusions can be proven rigorously, but only using techniques developed in the 19th century, which is at least a century after Grandi's death.
Here's the real take away from all of this. If you hear of a ridiculous mathematical claim or result from somebody, and if you are really that interested in it, then you should look into it a bit. While there have been plenty of hacks, most of them had no real impact on the development of mathematics and have been forgotten. If we actually remember somebody, then they probably did good work and you haven't heard the whole story yet.