# 1 + 2 + 3 + 4 + ⋯

## Divergent series / From Wikipedia, the free encyclopedia

#### Dear Wikiwand AI, let's keep it short by simply answering these key questions:

Can you list the top facts and stats about 1 + 2 + 3 + 4 + ⋯?

Summarize this article for a 10 years old

The infinite series whose terms are the natural numbers **1 + 2 + 3 + 4 + ⋯** is a divergent series. The *n*th partial sum of the series is the triangular number

- $\sum _{k=1}^{n}k={\frac {n(n+1)}{2}},$

which increases without bound as *n* goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.

Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −+1/12, which is expressed by a famous formula:[2]

- $1+2+3+4+\cdots =-{\frac {1}{12}},$

where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.[3]

In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".[4]

Oops something went wrong: