课程名称︰微积分甲下 课程性质︰必修 课程教师︰周青松 开课学院：管理学院、生农学院、理学院 开课系所︰工管系科管组、生工系、生机系、地质系 考试日期（年月日）︰2012/06/18 考试时限（分钟）：110分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : I. A. (10%) Show that ∫ dx/x^p convergent if p > 1 and diverges if 0＜p≦1. B. (10%) Let p be a real number.Show that Σ 1/k^p converges if p>1 and diverges otherwise. II. Let Σak be a series with non-negative terms, and suppose that (ak)^(1/k) →ρ.Show that A.(10%)If ρ<1, then Σak converges. B.(10%)If ρ>1, then Σak diverges. III. A.(10%)Expand g(x)=e^(x/2) in powers of x-3. B.(10%)Prove that 1 1 1 ln x = ln a + — (x-a) - －－(x-a)^2 + －－(x-a)^3 - ... a 2a^2 3a^3 for 0<x≦2a. IV. A.(10%)Use L'Hospital rule to evaluate the limit e^x-1-x lim －－－－－－. x→0 x arctan x B.(10%)Find a power series representation for the improper integral x ∫ arctan t/t dt. 0 V.Set f(x)=xe^x A.(10%)Expand f(x) in a power series. B.(10%)Integrate the series and show that ∞ 1 1 Σ －－－－－ = －－ n=1 n!(n+2) 2 --

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