課程名稱︰微積分甲下(暑修)
課程性質︰
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰2010/08/25
考試時限（分鐘）：120分鐘 (8:10~10:10)
是否需發放獎勵金：yes, thanks
試題 :
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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