<Dd> ∑ k = 1 n k = n (n + 1) 2, (\ displaystyle \ sum _ (k = 1) ^ (n) k = (\ frac (n (n + 1)) (2)),) </Dd> <P> 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 . </P> <P> 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, some of which have applications in other fields such as complex analysis, quantum field theory, and string theory . 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: </P> <Dl> <Dd> 1 + 2 + 3 + 4 + ⋯ = − 1 12, (\ displaystyle 1 + 2 + 3 + 4 + \ cdots = - (\ frac (1) (12)),) </Dd> </Dl>

What is the sum of all real numbers