Divergent series
BGN hyperreal summation

This summation method works by using an extension to the real numbers known as the hyperreal numbers. Since the hyperreal numbers include distinct infinite values, these numbers can be used to represent the values of divergent series. The key method is to designate a particular infinite value that is being summed, usually ω {\displaystyle \omega }, which is used as a unit of infinity. Instead of summing to an arbitrary infinity (as is typically done with ∞ {\displaystyle \infty }), the BGN method sums to the specific hyperreal infinite value labeled ω {\displaystyle \omega }. Therefore, the summations are of the form

This allows the usage of standard formulas for finite series such as arithmetic progressions in an infinite context. For instance, using this method, the sum of the progression 1 + 2 + 3 + … {\displaystyle 1+2+3+\ldots } is ω 2 2 + ω 2 {\displaystyle {\frac {\omega ^{2}}{2}}+{\frac {\omega }{2}}}, or, using just the most significant infinite hyperreal part, ω 2 2 {\displaystyle {\frac {\omega ^{2}}{2}}}.
oulipo:avoid_letter[t]