課程名稱︰微積分乙上 課程性質︰必帶 課程教師︰王振男 開課學院：醫學院 開課系所︰醫學系 考試日期（年月日）︰2011／1／11 考試時限（分鐘）：140min 是否需發放獎勵金：是！！ （如未明確表示，則不予發放） √(x-2) 1. Evaluate ∫──── dx (10%) √(x-1) 2. Find ∫√x(e^√x)dx (10%) ∞ 3. Give the series Σ[(-1)^n]arctan(1/n) n=1 Does it converge absolutely, converge conditionally, or diverge? (10%) 4. Find the interval of convergence of the following series (20%) ∞ Σ[1-cos(1/n)]x^n n=1 5. An executive conference room of a corporation contains 4500 ft^3 of air initially free of carbon monoxide. Starting at time t=0, cigarette smoke containing 4% carbon monoxide is blown into the room at rate of 0.3 ft^3/min. A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of 0.3 ft^3/min. Find the time when the concentration of carbon monoxide in the room reaches 0.01%. (20%) 6. Prove or disprove the following statements. ∞ a_n+1 ∞ (a) For a series Σa_n, if lim │───│＞1, then Σa_n diverges.(10%) n=1 n→∞ a_n n=1 ∞ ∞ (b) Let Σa_n converge, then Σ│a_n│^p converges for any p＞1.(5%) n=1 n=1 ∞ ∞ 7. A sequence {b_k} is called a subsequence of {a_n} if b_k = a_(n_k) k=1 n=1 for k = 1,2,…, where n_1＜n_2＜n_3＜…. Assume that lim a_n = 0. n→∞ ∞ ∞ Then there exists a subsequence {b_k} of {a_k} such that the series n=1 n=1 ∞ Σb_k converges absolutely. (15%) k=1 -- --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.7.59 ※ 編輯: shokanshorin 來自: 140.112.7.59 (01/11 15:07)