<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> <Dd> 1 + 2 + 3 + 4 + ⋯ = − 1 12, (\ displaystyle 1 + 2 + 3 + 4 + \ cdots = - (\ frac (1) (12)),) </Dd> <P> 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 . </P>

The sum of all positive integers less than or equal to n