Bk is a Bernoulli number. Integral function means If J(x) = Σpnxn is an integral function, then the J sum of the series a0 + ... is defined to be if this limit exists. There is a variation of this method where the series for J has a finite radius of convergence r and diverges at x = r. In this case one defines the sum as above, except taking the limit as x tends to r rather than infinity. Borel summation In the special case when J(x) = ex this gives one (weak) form of Borel summation. Valiron's method Valiron's method is a generalization of Borel summation to certain more general integral functions J. Valiron showed that under certain conditions it is equivalent to defining the sum of a series as where H is the second derivative of G and c(n) = e−G(n), and a0 + ... + ah is to be interpreted as 0 when h < 0. Moment methods Suppose that dμ is a measure on the real line such that all the moments are finite. If a0 + a1 + ... is a series such that converges for all x in the support of μ, then the (dμ) sum of the series is defined to be the value of the integral if it is defined. (If the numbers μn increase too rapidly then they do not uniquely determine the measure μ.) Borel summation For example, if dμ = e−x dx for positive x and 0 for negative x then μn = n!, and this gives one version of Borel summation, where the value of a sum is given by There is a generalization of this depending on a variable α, called the (B′,α) sum, where the sum of a series a0 + ... is defined to be if this integral exists. A further generalization is to replace the sum under the integral by its analytic continuation from small t. Miscellaneous methods 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 , which is used as a unit of infinity. Instead of summing to an arbitrary infinity (as is typically done