課程名稱︰微積分甲下
課程性質︰署修
課程教師︰周青松
開課系所︰數學系
考試時間︰2006/9/6
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試題 :
I.
(A) Prove that,
∞ k 1
(i) if |x| < 1, then Σ x = ───
n=0 1-x
∞ k
(ii) if |x| ≧ 1, then Σ x diverges.
n=0
∞ k-1 1
(B) Show that Σ kx = ──── , for |x| < 1
k=0 (1-x)^2
II.
∞ n -αk
(A) Prove that, Σ k e
k=0
converges for each nonnegative integer n and α > 0.
(B) Use the integral test to show that
∞ 1
Σ ─── converges for p > 1
k=1 k^p
III.
(A) Prove that
∞ (-1)^k 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)!
IV.
Find a power series representation for the improper integral
x ln(1+t)
(A) ∫ ──── dt
0 t
x sinh(t)
(B) ∫ ──── dt
0 t
V.
(A) For each integer n and all γ≠0, where γ=xi+yj+zk and r =║γ║.
Prove that
n n-2
▽r =(nr )γ
(B) Find the directional derivative of the function
2
f(x,y,z) = 2xz cos(πy)
at the point P(1,2,-1) toward the point Q(2,1,3)
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