= 0 + 0 + 0 + ...
  = (1 - 1) + (1 - 1) + (1 - 1) + ...
  = 1 - 1 + 1 - 1 + 1 - 1 + ...
  = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
  = 1 + 0 + 0 + 0 + ...
  = 1

Ubaldus, a Catholic Saint, felt that this proved the existence of God because "something has been created out of nothing."

Actually, I think it proves that his math is all jacked up because Ubaldus is treating an infinite series like a finite one. Line 3 is the illegal operation. It is patently false when dealing with ellipses. The dots matter if you are going to perform operations for which the last number becomes important. Once he grouped the numbers parenthetically in line 3, the statement is only true if the series ends with a -1, so he uses a classic bit of polemic by changing definitions and counting on you to miss the substitution. Math proofs work for math and often predict things in the real world, but this is only useful to the extent that the conclusions can be tested.

I think the step from line 4 to line 5 here is almost the dodgiest.

It asserts that the series
-1+1-1+1... = sum((-1)^n)
converges to a sum of 0.

Even a high school kid should know that the alternating series diverges.

In my opinion, worse still is that this Ubaldus guy believes that his false proof of 0 = 1 means that god must exist! If anything, it should only prove that 0 = 1 and mathematics = rubbish.

Wait a minute, couldn't one use the exact same reasoning to prove the non-existance of god? A false statement implies anything. It's like asking what would happen if something impossible happened, it is complete rubbish.

You think that's weird? Try this:
1 =  = x = 1-1+1-1+1-1+... =
= 1-(1-1+1-1+1-1+...) =
1-x
so x, in addition to being 0 and 1, is also 1-x. High school algebra yields x=1/2.

The point of all this? You cannot parenthesise the elements of an infinite series if it is divergent, and expect results that make sense. You cannot parenthesise the elements of an infinite series that is not absolutely convergent and expect results that make sense (see Riemann's theorem on parenthesising infinite series that converge but not absolutely for even more amazing stuff).

Caution is warranted when doing calculus. Seemingly innocuous expressions may turn out to be meaningless (or to mean something other than what you think they do).

The problem is this:

Ubaldus made an algebraic mistake.

In lines 2 and 3, he included 3 +1's and 3 -1's. All's fine and well and everything equals zero. In line 4, there are 4 +1's and 3 -1's.

You can't add a term to one side without adding it to another

In essence, Ubaldus added 1 to the right side without completing the same operation on the left He may have been dealing with infinite series, but zero is zero anyway you slice it, even if you're talking about a billion terms of (1-1). The number of terms does not affect the outcome of the operation, until a person erroneously adds a 1 without adding a -1. Of course this will make falsely suggest 0 = 1. Had this worked out, I am not sure it proves there is a God anyway..
Quoth my wife, the math major:

"Very cute. They show us this in calculus as "reasons to be careful with infinite series," but I've never seen it given as a proof of the existence of god... exactly how does it prove the existence of god, anyway? You'd think it'd prove that you can get something from nothing without god."

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.

Cheers!

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