课程名称︰微积分甲上 课程性质︰必修 课程教师︰王金龙 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010/12/30 考试时限（分钟）：30 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : ∞ (k!)^2˙x^2 A. Determine whether the series Σ ────── k=1 (2k)! converges or diverges for all x ε R (ε：属於) B. Let f(x) = x^2 for x ε [-π,π] and f( x + 2π) = f(x) for all x. (a) Find the Fourier series of f. ∞ 1 π^2 (b) Use (a) to show that Σ ─── = ──. k=1 k^2 6 π sin(n+1/2)t C. (a) Show that ∫ ─────── dt =π for all n ε N. 0 sin(t/2) 1 (b) Suppose that f is piecewise C on [ a , b ], show that b lim ∫ f(t)sinλt dt = 0. λ→∞ a --

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