[考古] [97下][期中考][微積分(二)][李英豪][먠…
[開課學院]: 工學院(工學院只能修他的微積分綜合班)
[開課系所]: 綜合班..
[課程名稱]: 微積分(二)
[老師名稱]: 李英豪老師
[開課學期]: 97-2
[類型]: 97年第二學期期中考
(一)是非題
(a) lim(1-1/n)^n = e^(-1)
n→∞
(b) lim n^(300/n) = 1
n→∞
∞ ∞
(c)If Σan converges, then the series Σ(an)^2 also converges
n=1 n=1
∞
(d)The p-series Σ1/n^(1.001) is converges
n=1
∞
(e)The geometric series Σ 2^(2n)˙5^(1-n) is ocnverges
n=1
∞
(f)The series Σ (-1)^(n+1)˙(n+1)/n is ocnverges
n=1 (乘)
_
(g)cos^(-1)(－√3 / 2 ) = 5π/ 6
↑ (意為反三角函數 Acos..)
(h)sin^(-1)[sin(2π/3)] = 2π/3
(i)lim x-tanx / x^3 = 1/6
x→0
____________
(j)lim(√x^2 + 3x - x - x) = 3/2
(二)填充題
∞
1.suppose that we want to find the sum of the telescoping series Σ 1/(n(n+1))
n=1
∞
(a)For n≧1,the nth partial sum Sn of the series is Sn = Σ 1/(k(k+1)) = ______
n=1
∞
(b)The sum of the series Σ 1/(n(n+1)) is S =lim Sn = _______
n=1 n→∞
0
2.suppose that we want to find the impropeer integral ∫ x˙e^x dx
-∞
0 0
By definition,we known that ∫ x˙e^x dx =lim ∫ x˙e^x dx
-∞ -∞
0
(a) For t < 0,∫ x˙e^x dx = _______
-∞
(b) lim t˙e^t = _______
t→-∞
0
(c)∫ x˙e^x dx = _______
-∞
∞
3.The sum of the Series Σ 3 /(n(n+1)) + 5 / (2^n) is S = _______
n=2
4. (a) d/dx tan^(-1)(㏑x)=________ (b) tan[sin^(-1)(1/3)] = ________
∞
(b) lim tan^(-1)t = ________ (d) ∫ 1 / (π˙(1 + x^2)) dx = _______
t→-∞ -∞
(三) 計算與證明
1. Find the following integrals
_______
(a) ∫e^x ˙ sin x dx (b) ∫ 1/(x^2 ˙ √x^2 + 4 ) dx
3 _ ______
(c) ∫ 1/ √x dx (d) ∫ 1 / (x ˙√x - 1) dx
0
2. Find the following limits
(a) lim x^x (b)lim (㏑n)/ n
x→0 n→∞
3. Determine whether the following series is convergent or divergent?
∞ ∞ ∞ ∞ _
(a)Σ n/(n+1) (b)Σ 1 /(n^2 - 1) (c)Σ (-3)^n / n! (d)Σ (-1)^n ˙ 1 / √n
n=1 n=1 n=1 n=1
Hint: (a) By divergence test (b) By limiting comparison test
(c) By Ratio test (c) By alternating series test
∞
4.(a)If Σ an converges, show that lim an =0
n=1 n→∞
(b)Give an example to show that the statement
∞
"If lim an = 0, then the series Σ an converges is false".
n→∞ n=1
以上...........我以後都不打數學的東西了....好難打阿= =...
如果有看不懂，歡迎來信問我......
這種東西應該貼圖比較理解..可是這樣會有獎金嗎XD
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