作者bookh (book)
看板NTU-Exam
标题[试题] 98暑 周青松 微积分甲下 期中考
时间Wed Sep 8 10:08:08 2010
课程名称︰微积分甲下(暑修)
课程性质︰
课程教师︰周青松
开课学院：
开课系所︰
考试日期（年月日）︰2010/08/25
考试时限（分钟）：120分钟 (8:10~10:10)
是否需发放奖励金：yes, thanks
试题 :
It's necessary to explain all the reasons in detail and show all of your work
on the answer sheet. Or you will NOT get any credits. If you used any theorems
in textbook or proved in class, state it carefully and explicitly.
1.(a) Find lim (1+x)^(1/x)
x→0+
(b)Determine whether the sequence (x^100n)/n! converges as n→∞
If it dose, find the limit of the sequence.
2.(a) For what values of r is
∞
∫ x^r e^(-x) dx
0
convergent?
∞
(b) Show by induction that ∫ x^n e^-x dx=n! , n=1,2,3...
0
3.(a) Show that
k
∞ (-1) 2k
cos x = Σ ———— x for all real x
k=0 (2k)!
(b) Show that
∞ 1 2k
cosh x = Σ ———— x for all real x
k=0 (2k)!
k+1
∞ (-1) k
4.(a) Show that ln (1+x)=Σ ———— x for all -1<x≦1
k=1 k
k+1
∞ (-1) 2k-1
(b) Show that arctan x =Σ ———— x for all -1≦x≦1
k=1 2k-1
5.Set f(x)=xe^x
(a) Expand f(x) in a power series
(b) Integrate the series and show that
∞ 1 1
Σ ———— = —
n=1 n!(n+2) 2
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