1 + 2 + 3 + 4 + ⋯

Uncommon card · No. 104157

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...

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