The famous mathematician Ramanujan provided us with numerous extraordinary formulas. Today, I'll explore one that appears straightforward but reveals a highly counterintuitive result:

$$

\sum_{n=1}^\infty n = -\frac{1}{12}

$$

Initially, this conclusion seems quite perplexing. Even pupils understand that the sum of all natural numbers should be infinite, so how could it result in a negative number, let alone a fraction? Before rushing to conclusions, it's important to note that mathematical problems involving infinity often defy intuition. Let's examine the proof below.

## Schrödinger's Cat

Schrödinger's cat is a famous thought experiment in quantum mechanics proposed by Erwin Schrödinger in 1935. It illustrates the concept of quantum superposition. In the experiment, a cat is placed in a sealed box with a radioactive atom, a Geiger counter, a vial of poison, and a hammer. If the atom decays, the Geiger counter triggers the hammer to break the vial, releasing the poison and killing the cat. According to quantum mechanics, until the box is opened and the system is observed, the cat can be considered both alive and dead simultaneously.

How is this connected to the mathematical problem we're trying to solve? Let's look at the following summation first:

$$

S_1 = 1-1+1-1+1-1+1-1 \dots

$$

What's the outcome of this summation? Finding a precise answer seems challenging. If we take even terms, the result is $0$; if we take odd terms, the result is $1$. This resembles the cat that is simultaneously dead and alive, doesn't it? Since it's in a superposition of $0$ and $1$, mathematicians take a cue from quantum mechanics, employing probability to determine the summation's result. We know that even and odd numbers each represent half of the natural numbers, so the result is $S_1 = \frac{1}{2}$.

## Building the Proof

Next, let's establish the second summation

$$

S_2 = 1-2+3-4+5-6+7-8 \dots

$$

By subtracting $S_1$ from $S_2$, we obtain

$$

\begin{aligned}

S_2-S_1 &= -1+2-3+4-5+6-7+8 \dots \\

&= -(1-2+3-4+5-6+7-8 \dots) \\

&= -S_2

\end{aligned}

$$

So, $S_2 = \frac{S_1}{2} = \frac{1}{4}$.

Let's define $S_3 = 1+2+3+4+\dots = \sum_{n=1}^\infty n$, then

$$

\begin{aligned}

S_3-S_2 &= 4+8+12+16+ \dots \\

&= 4(1+2+3+4+ \dots) \\

&= 4S_3

\end{aligned}

$$

So, $S_3 = -\frac{S_2}{3} = -\frac{1}{12}$.

$\blacksquare$

## Practical Use in Physics

This seemingly absurd answer plays a significant role in physics.

In string theory, assuming the number of spatial dimensions is $d$, the minimum total energy of a string excited into a photon is

$$

E=\hbar\omega_1+(d+1)\sum \frac{1}{2}\hbar\omega_n

$$

Since $\omega_n=n\omega_1$, the total energy of this photon becomes

$$

E=\frac{1}{2}\hbar\omega_1\Big(2+(d-1)\sum_{n=1}^\infty n\Big)

$$

Here, $\sum_{n=1}^\infty n$ appears.

According to relativity, the lowest energy of a photon should be zero, so a string theory compatible with relativity must satisfy

$$

2+(d-1)\sum_{n=1}^\infty n = 0

$$

By substituting $\sum_{n=1}^\infty n=-\frac{1}{12}$, we can deduce $d=25$, indicating that there must be $25$ spatial dimensions. And with time included, there are a total of $26$ dimensions in spacetime.

In superstring theory, due to the introduction of supersymmetry factors, the ground state energy of the string increases by three times, and the photon energy constraint becomes

$$

2+3(d-1)\sum_{n=1}^\infty n = 0

$$

By substituting $\sum_{n=1}^\infty n=-\frac{1}{12}$, we find $d=9$. And adding one time dimension, a total of $10$ dimensions in spacetime are formed.

## One Incorrect Proof

An incorrect proof online claims that the sum of all natural numbers equals $-\frac{1}{8}$. Let's examine this flawed argument.

$$

\begin{aligned}

S &= 1+2+3+4+5+6+7+8+9+10+\dots \\

&= 1+(2+3+4)+(5+6+7)+(8+9+10)+\dots \\

&= 1+9+18+27+36+\dots \\

&= 1+9(1+2+3+4+\dots) \\

&= 1+9S

\end{aligned}

$$

Thus, $-8S = 1, S = -\frac{1}{8}$.

The error here is that **the associative law generally does not apply to summing infinite series**.

When dealing with infinite series, using the associative law can lead to problems. This is due to the infinite number of terms in such series.Although the associative law guarantees that the order of summation won't change the outcome for finite series, with infinite series, altering the order can result in divergence or convergence to varying values.