with ), the BGN method sums to the specific hyperreal infinite value labeled . 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 is , or, using just the most significant infinite hyperreal part, . Hausdorff transformations . Hölder summation Hutton's method In 1812 Hutton introduced a method of summing divergent series by starting with the sequence of partial sums, and repeatedly applying the operation of replacing a sequence s0, s1, ... by the sequence of averages , , ..., and then taking the limit . Ingham summability The series a1 + ... is called Ingham summable to s if Albert Ingham showed that if δ is any positive number then (C,−δ) (Cesàro) summability implies Ingham summability, and Ingham summability implies (C,δ) summability . Lambert summability The series a1 + ... is called Lambert summable to s if If a series is (C,k) (Cesàro) summable for any k then it is Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value . Le Roy summation The series a0 + ... is called Le Roy summable to s if Mittag-Leffler summation The series a0 + ... is called Mittag-Leffler (M) summable to s if Ramanujan summation Ramanujan summation is a method of assigning a value to divergent series used by Ramanujan and based on the Euler–Maclaurin summation formula. The Ramanujan sum of a series f(0) + f(1) + ... depends not only on the values of f at integers, but also on values of the function f at non-integral points, so it is not really a summation method in the sense of this article. Riemann summability The series a1 + ... is called (R,k) (or Riemann) summable to s if The series a1 + ... is called R2 summable to s if Riesz means If λn form an increasing sequence of real numbers and then the Riesz (R,λ,κ) sum of the series a0 + ... is defined to be Vallée-Poussin summability The series a1 + ... is called VP (or Vallée-Poussin) summable to s if where is the gamma function. . See also Silverman–Toeplitz theorem Notes References . . . . . . . Werner Balser: "From Divergent Power Series to Analytic Functions",