1 + 2 + 3 + 4 + ... = -1/12

The infinite series $1 + 2 + 3 + 4 + \cdots$ is divergent, meaning that it does not converge to a finite value. However, the mathematician Srinivasa Ramanujan famously assigned it the value of $-1/12$.

This result is often cited in certain branches of theoretical physics such as bosonic string theory and quantum field theory. An example would be the calculation of the Casimir effect, where the energy of a vacuum is computed using a regularized version of a divergent series.

Consider the following manipulation of the series:

Next, we can recognize that the alternating series $1 - 2 + 3 - 4 + \cdots$ is the power series expansion of the function $\frac{1}{(1+x)^2}$ evaluated at $x=1$. This gives us:

Dividing both sides by $-3$ gives us $c = -\frac{1}{12}$.

The calculation above treats an infinite series as if it were a finite sum, which is not mathematically rigorous. With divergent series, rearranging items or inserting zeroes in different places can change the result or lead to contradictions.

To handle divergent series in a more rigorous way, mathematicians use techniques such as analytic continuation and zeta function regularization. These methods allow us to assign finite values to divergent series in a consistent manner, which is why the value of $-1/12$ for the series $1 + 2 + 3 + 4 + \cdots$ is accepted in certain contexts.