sol for exam 2 take home

# 6 m 2b give the taylor series expression of gx fx

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: expression of +gx fx Answer for (2b) ∞ : ζm −1 m m=2 2 ζ xm −1 = x 2 3 ζ −1 + 2 x 3 ζ −1 + x 5 6 ζ 5 −1 x 4 4 −1 + 4 + 3 5 ζ . x 6 6 + ··· . : Work for (2b) +gx n = lim − n− ∞ → fx k=2 = lim − n− ∞ → n k=2 1 x x − ln 1 − k 2 1 x− k n ln 1− k=2 1− x k x 3 · ··· · 1− x n . In this last quantity, substitute x with −x. Then multiply −1 with it. Then the outcome is −f x + g x . Namely: 7 f x +g x =− −f +g −x −x . Hence the answer for (2b) is simply − ∞ m=2 −1 m−1 m ζm −1 m −x , which is simpliﬁed as ∞ m=2 (3) ζm −1 m Deﬁne the Euler’s Gamma function Γ x xm . as follows: +∞ Γx tx−1 e−t dt = x>0 t=0 Clearly, this function Γ x . is a generalization of the notion of factorials, in the following sense: For a non-negative integer m, Γ m+1 = m...
View Full Document

## This note was uploaded on 01/12/2014 for the course MATH 116 taught by Professor Scholle,minho during the Fall '08 term at Kansas.

Ask a homework question - tutors are online