sol for exam 2 take home

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

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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...
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