課程名稱︰微積分甲上 課程性質︰必修 課程教師︰康明昌 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰2011/01/14 考試時限（分鐘）：150(15:34~18:04) 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 滿分135 ∞ 1.(15%)Let a_1≧a_2≧．．．≧a_n≧．．．＞0 Show that Σa_n converges if ∞ n=1 and only if Σ(2^k．a_2^k) converges k=0 2.(15%)For -1<x<1, consider the Taylor's expansion ∞ arc sinx/(1-x)^0.5 = Σ(a_n．x^n). Find a_1 a_2 a_3 a_4 n=0 3.(15%)Evaluate lim x^2[(1+1/x)^x-e㏑(1+1/x)^x] x→∞ 4.(15%)Find the area of the loop bounded by the folium of Descartes x^3+y^3 = 3axy where a>0 (請GOOGLE圖片 folium of descartes) 5.(15%)Let γ=f(θ) be the equation of a curve in polar coordinates Show that the curvature κ is given by the formula κ = (2γ'^2-γγ"+γ^2)/(γ'^2+γ^2)^1.5 where γ'=df/dθ γ"=d^2f/dθ^2 6.(15%)Let 0<ε<N Show that ∞ Σ[(1/2n+1)．(x-1/x+1)^(2n+1)] converges uniformly for ε≦x≦N n=0 7.Let α be any real number. Define f_n(x) by ╓n^α．x if 0≦x≦1/n ║ f_n(x)=╟n^α(2/n - x) if 1/n≦x≦2/n (※n≧2) ║ ╚0 if 2/n≦x≦1 (i)(15%)For 0≦x≦1. Show that lim f_n(x) = 0 n→∞ (ii)(15%)Determine all the values α such that f_n(x)→0 as n→∞ uniformly for 0≦x≦1 ∞ 8.(15%)Let x_0≠0. Assume that Σ(a_k．X_0^k) converges. ∞ k=0 Show that Σa_k．x^k converges for all x satisfying 0≦|x|≦|x_0| k=0 --

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