Is the sum of all integers a negative number?

A warning to always be rigorous

Linguistics, social sciences, and mathematics are sciences. Not only due to their study of tangible objects Since they all study objects created by the emergent complexity of the human mind: language, the consequences of social interactions, and geometric-numeric abstractions (respectively.), but because of the rigor with which they guide themselves. However, when we allow ourselves to not be rigorous some curious results can occur. HereI originally wrote this document for a friend of mine, which I think is better suited to exist in this format., we’ll see one of these: the sum of all positive integers is $-\frac{1}{12}$.

Intro to Infinity

A series is the sum of a (numerable) infinite amount of terms, in particular we can sum infinitely $1$ and $-1$. Let’s call this sumIn Spanish there is a fervent debate between calling the sum of many terms either a _suma_ or a _sumatoria_. Both are valid ways, since any language is made by its speakers (and writers). $a$, then

\[a=-1+1-1+\cdots\]

At a first glance, one would think that such a sum does not exists, as it cannot be computedAnd one would be right. nonetheless we could try to use some high-school algebra to make the most out of the definition. For example:

\[\begin{aligned} a&=1-1+1-1+\cdots\\ 1-a&=1-(1-1+1-1+\cdots)\\ &=1-1+1-1+\cdots\\ 1-a&=a\\ 1&=2a\\ \end{aligned}\]

We then conclude that the sum $a=\frac{1}{2}$.

An interesting series

Math is built one result at a time. Hence, we can use this last result to deal with the new sum $b$

\[b=1-2+3-4+\cdots\]

Expanding upon $b$,

\[\begin{aligned} 2b&=b+b\\ &=(1-2+3-4+\cdots)+(1-2+3-4+\cdots)\\ &=1+(-2+3-4+\cdots)+1-2+(3-4+\cdots)\\ &=\cancel{1}+(-2+3-4+\cdots)+\cancel{1}-\cancel{2}+(3-4+\cdots)\\ &=(-2+3-4+\cdots)+(3-4+5-\cdots)\\ &=(-2+3)+(3-4)+(-4+5)+\cdots\\ &=1-1+1-1+\cdots\\ &=\frac{1}{2}\end{aligned}\]

We can see that $b=\frac{1}{4}$

Is the sum of the integers a negative number?

To conclude, let us pose the next question: What would happen if we summed the 1, then the 2, and so on until we are left out of numbers? Let’s name this sum $s$, defined as

\[s=1+2+3+\cdots\]

Then

\[\begin{aligned} s&=1+\textcolor{Red}{2}+3+\textcolor{Red}{4}+5+\textcolor{Red}{6}\cdots\\ 4s&=\textcolor{Red}{4}+\textcolor{Red}{8}+\textcolor{Red}{12}\\ s-4s&=1-2+3-4+5-6\\ -3s&=\frac{1}{4}\\ s&=-\frac{1}{12} \end{aligned}\]

In the end, when we “sum” all the positive integers we end up with a negative fraction! I leave it as an exercise to the reader to examine if this result means anything outside the paper (or this blog.)