Question

What's the intuition behind the equation 1+2+3+⋯=−1121+2+3+⋯=−112 ?

Answer 1

The sum clearly diverges. This is indisputable. The point of the claim above, that

$1+2+3+\cdots=-\dfrac1{12}$

is to demonstrate that a sum of infinitely many terms can be manipulated in a variety of ways to end up with a contradictory result. It's an artifact of trying to do computations with an infinite number of terms.

The mathematician Srinivasa Ramanujan famously demonstrated the above as follows: Suppose the series converges to some constant, call it $C$. Then

$\begin{matrix}C&=&1&+2&+3&+4&+5&+6&+\cdots\\4C&=&&+4&&+8&&+12&+\cdots\\-3C&=&1&-2&+3&-4&+5&-6&+\cdots\end{matrix}$

Now, recall the geometric power series

$\displaystyle\sum_{n\ge0}x^n=1+x+x^2+x^3+\cdots=\dfrac1{1-x}$

which holds for any $|x|$. It has derivative

$\displaystyle\sum_{n\ge1}nx^{n-1}=1+2x+3x^2+4x^3+\cdots=\dfrac1{(1-x)^2}$

Taking $x=-1$, we end up with

$1+2(-1)+3(-1)^2+4(-1)^3+\cdots=1-2+3-4+\cdots=\dfrac14$

and so

$-3C=\dfrac14\implies C=-\dfrac1{12}$

But as mentioned above, neither power series converges unless $|x|$. What Ramanujan did was to consider the sum $1-2+3-4+\cdots$ as a limit of the power series evaluated at $x=-1$:

$\displaystyle-3C=\lim_{x\to-1^+}\sum_{n\ge1}nx^{n-1}=\lim_{x\to-1^+}\frac1{(1-x)^2}=\frac14$

then arrived at the conclusion that $C=-\dfrac1{12}$.

But again, let's emphasize that this result is patently wrong, and only serves to demonstrate that one can't manipulate a sum of infinitely many terms like one would a sum of a finite number of terms.