課程名稱︰微積分甲下
課程性質︰必修
課程教師︰周青松
開課學院：理學院
開課系所︰地科系 生工系 生機系 工管系科管組
考試日期（年月日）︰104/4/27
考試時限（分鐘）：110分鐘
試題 :
(I) (20 pts)
(A) Show that lim (1+1/n)^n =e
n→∞
(B) Show that for each real x, lim (1+x/n)^n =e^x
n→∞
(II) (20 pts)
(A) Determine whether the improper integral∫∞ r/(r^2+x^2) dx (r>0)
-∞
converges or diverges. If it converges, give the value of it.
(B) Evaluate∫1 1/ x^(4/5) dx
-2
(III) (20 pts)
Determine whether the following series converge or diverge.
∞ 2^k ∞ k^k
(A) Σ ----------- (B) Σ ---------
k=1 k^3 k=1 k!
(IV) (20 pts)
∞ (-1)^k
(A) Show that for all real x, cosx =Σ ---------- x^(2k)
k=0 (2k)!
∞ (-1)^k
(B) Show that for all│x│<1, ln(1+x)=Σ---------- x^(k+1)
k=0 k+1
(V) (20 pts)
e^x-1-x
(A) Evaluate lim －－－－－－
x→0 x arctan x
by using (i)L'Hopital's rule (ii)power series.
(B) Deduce the differentiation formulas,
d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x,
from the expansions of sinh x and cosh x in powers of x.
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