### Introduction One of the more confusing aspects of mathematics is diverging series. One famous example is the following sum: 1 + 2 + 3 + 4 + 5 + ... This sum obviously infinite. However, in advanced mathematics textbooks one will instead find that the sum is 1 + 2 + 3 + 4 + 5 + ... = - 1 / 12 ! In fact, this result is used in string theory (and is the reason why bosonic string theory can only exist in 26 dimensions). This result is bizzare - ignoring the fact that the sum is supposedly finite, it is even a negative number! http://www.crystalinks.com/infinitysymbolbw.jpg This example is a well known one, which we won't get into at the moment (although I am willing to make a post explaining this sum too if there's enough interest). Instead, we'll be concentrating on a different sum - [Grandi's series](https://en.wikipedia.org/wiki/Grandi's_series), which is defined as 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + .... Grandi studied this sum, and actually used it in order to show that god exists! Let's follow his argument to see if we are convinced. ### Proof that god exists We begin by trying to calculate Grandi's sum naively, using brackets: 1 - 1 + 1 - 1 + 1 - 1 + .... = ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + .... = 0 + 0 + 0 + ... = 0 we thus see that Grandi's sum is zero. On the other hand, we can calculate the sum using brackets in a different order: 1 - 1 + 1 - 1 + 1 - 1 + .... = 1 + ( - 1 + 1) + ( - 1 + 1) + ... = 1+ 0 + 0 + 0 + ... = 1 And we have thus proven that 0 = 1 ! We have thus found that "something" can be created out of "nothing", since 0 and 1 are equal. This argument was used by Grandi to show that God does exist, since this "proves" that the story of creation from the book of Genesis is correct! God was there to create something (the number 1) out of nothing (the number 0) - "let there be light"! ### Wait.... what? While amusing, this is definitely not a proof of the existence of god. But even so, how do we explain the apparent paradox, where we proved that 0 = 1 ? The answer is that our starting point was completely wrong. When we think of a diverging sum, we usually think of a sum which is infinite (like 1 + 2 + 3 + ...). But mathematically, another type of diverging sum is one whose sum cannot be defined. The perfect example is Grandi's sum, since summing more and more numbers won't make the sum converge to any one number. Thus the sum 1 - 1 + 1 - 1 + ... is diverging, meaning that mathematically, it **does not exist**. Thus, any conclusions we make from it are completely meaningless. (Specifically, it was our use of brackets in a diverging series that caused our downfall) This proof that 0 = 1 is just like all of the other proofs that exist (like [this one](https://www.math.hmc.edu/funfacts/ffiles/10001.1-8.shtml)), where one assumes something that is completely wrong and then proceeds to find nonsensical results. Nevertheless, its historical background is still pretty amusing!