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◆ From: 140.112.7.59
※ 編輯: shokanshorin 來自: 140.112.7.59 (01/11 15:07)
推 ALegmontnick:done 01/11 15:38
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
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