A couple days ago, my friend Adolfo posted a rant on Facebook about the famous (or infamous) series
$$\sum_{n=1}^\infty n=-\frac{1}{12}\,.$$
His comment was not about the result itself, but about the little effort made by mathematicians to clarify this voodoo.
This result is very counter-intuitive as it states that if you add all positive integers, you will not only get a negative quantity, but also a fraction! I remember speaking about this result a couple years ago with a friend who was skeptical about it and he was telling me that it shouldn't be true since if he grabbed a calculator and start adding 1+2+3+4+..., he wouldn't approach a negative number, on the contrary, it would become bigger and bigger!
The trick lies in realizing we are talking about a mathematical object called a
series, which is essentially different that any usual
finite sum. One thing that can help to see that series are different than just adding a finite number of terms is that sometimes the order in which we add makes a difference! This is called
Riemann rearrangement theorem, and states that if a series is
conditionally convergent, then we can permute its terms to make it
converge to any real number, or to make it
diverge.
There are several concepts that play important roles here. One of them is the idea of dealing with
an infinite number of things. It is not natural to operate with an infinite number of objects and a major problem is that we cannot directly apply an
algorithm to compute this, as one of the key things of algorithms is that they
must end, in other words, they must have a finite number of steps.
Hence, the traditional addition algorithm that we use in everyday life (or at least that our calculators use), cannot be applied directly to infinite series. Then the relevant question regarding
$$\sum_{n=1}^\infty n$$
is not how much is it? but rather, what is it?
The traditional approach for
computing infinite series is by means of
sequences of partial sums. This means that since we don't know what does it mean to add an infinite number of things, we approach infinity in the
potential infinity sense, which establishes that
infinity is the ability of taking an increasing sequence of big numbers forever. In other words, we take a big number of terms, we add them, and we assume that this result should somehow be close to the
value of the series. Then we take a bigger amount of terms, and a bigger, and we keep doing this. Eventually, if the results obtained from the finite sums tend to cluster around a value, we say that that value is the result of the infinite series.
It is important to remark that when we write
$$\sum_{n=1}^\infty\frac{1}{2^n}=1\,,$$
we really don't mean equal in the same sense we mean 1+1=2. This first equal sign really means
$$\lim_{k\to\infty} s_i=1\,,$$
where
$$ s_i=\frac{1}{2}+\frac{1}{4}+\dots+\frac{1}{2^i}\,.$$
Of course writing $\sum 1/2^n=1$ is shorter and conveys the same idea, or at least that's what lazy mathematicians think. Again, the equal sign used here should not be thought as a comparison between two objects but rather as assigning a value to the series.
This assignment is very intuitive and we could say is very natural to think that this is a reasonable way to approach defining the meaning of an infinite series. But this it's not the only option. Deeper questions arise when this method of partial sums don't provide an answer, as it is the example of $\sum n$. The easy way out is simply to say there is no answer and the series diverges. But just as it happened with the equation $x^2=-1$, mathematicians saw an opportunity to extend the theory and also assign values to series for which the partial sums method is not enough.
It is possible to use the idea of
actual infinity instead of potential infinity in order to approach these objects. An actual infinity approach considers having
all infinite terms at the same time, as opposed to just a never-ending source of terms. Euler and Leibniz first started to develop ideas around
divergent series and one of their key insights was to
look at the meaning of a sum rather than its value.
A very interesting example of this occurs with the series
$$\sum_{n=0}^\infty (-1)^n\,.$$
This is an alternating sum for which the traditional approach of partial sums gives no answer (the partial sums jump from 0 to 1 and back forever). But if we look a more abstract realization of the series, we can make it correspond to a geometric series,
$$\sum_{n=0}^\infty r^n\,.$$
This is another representation of the function $1/(1-r)$ inside the unit circle. With this we can make the conceptual assignment of $r=-1$ to this series and make the correspondence
$$\sum_{n=0}^\infty (-1)^n\mapsto \frac{1}{2}\,.$$
Sometimes we abuse of the notation and we prefer to write it down as $\sum(-1)^n=1/2$, but we have to stress the fact that "=" does not represent a comparison between objects but rather a correspondence.
Something similar happens with the series $\sum n$. One of the most used methods involve the famous
Riemann Zeta function. This is defined as
$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}\,,$$
for complex numbers $s$ with $\Re(s)$ bigger than 1. It is possible to
extend the definition of the Zeta function to the entire complex plane using the
reflection formula,
$$\frac{\zeta(1-s)}{\zeta(s)}=\frac{2\Gamma(s)}{(2\pi)^s}\cos\left(\frac{\pi s}{2}\right)\,.$$
Using the series representation of the Zeta function suggest the correspondence
$$\sum_{n=1}^\infty n\mapsto\zeta(-1)\,.$$
Hence, we can assign the
value of
$$\zeta(-1)=\zeta(2)\frac{2\Gamma(s)}{(2\pi)^s}\cos\left(\frac{\pi s}{2}\right)=-\frac{1}{12}$$
to our series. Assign, not evaluate.
