Asymptotic Behaviors of Intermediate Points in the Remainder of the Euler-Maclaurin Formula

and Applied Analysis 3 in an infinite number of variables x1, x2, . . ., defined by the series expansion 1 k! (∑ m≥1 xm t m! )k ∑ n≥k Bn,k t n! , k 0, 1, 2, . . . . 2.2 Their explicit expressions are given by the formula Bn,k x1, x2, . . . , xn−k 1 ∑ n! a1! 1! a2! 2! a2 · · · x1 1x2 a2 . . . , 2.3 where the summation takes place over all nonnegative integers a1, a2, . . ., such that a1 2a2 · · · n and a1 a2 · · · k. For example, we have B0,0 1, B1,1 x1, B2,1 x2, B2,2 x1, B3,1 x3, B3,2 3x1x2, B3,3 x1, . . . , Bn,1 xn, Bn,n x1. 2.4 For more important properties the reader is referred to 15 . 3. Asymptotic Expansions of Intermediate Points In this section, we will consider asymptotic behavior of the point ξ in 1.4 . Before the main result is given we first present an essential lemma. Lemma 3.1 see 15 . The following identity: