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, 1 2 3 infinity pdf 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 a monograph on moonshine theory, Terry Gannon calls this equation “one of the most remarkable formulae in science”.

This equation was known to the Pythagoreans as early as the sixth century BCE. Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.