1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the positive integers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number ∑ k = 1 n k = n ( n + 1 ) 2 , {\displaystyle \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. While the series itself diverges to infinity, in certain mathematical contexts it can be assigned a finite value. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of ⁠−+1/12⁠, which is expressed by the famous formula 1 + 2 + 3 + 4 + ⋯ = − 1 12 , {\displaystyle 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 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. In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".