課程名稱︰微積分甲上
課程性質︰數學系大一必帶
課程教師︰陳榮凱
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013/10/17
考試時限（分鐘）：30
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
The total is 25 points.
1 1 １ 1 1 １
(1) (12 pts) Let a = - + --- + ... + -- and let b = --- + --- + ... + --.
n n n+1 2n n n+1 n+2 2n
(a) Prove that both sequence {an} and {bn} converge.
(b) Prove that they have the same limits.
(2) (13 pts) Let f(x) be a continuous function in the interval [a,b]. Suppose
that f(x)>=0 for all xε[a,b] and f(x0)>0 for some a<x0<b.
(a) Show that there exists an interval [x0-δ,x0+δ] C [a,b] so that f(x)
>=f(x0)/2 for all xε[x0-δ,x0+δ].
b
(b) Prove that∫f(x)dx > 0.
a
//ε和C是屬於符號 打不出來請見諒 (-_-;)rz
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※ 編輯: xavier13540 來自: 140.112.196.111 (11/13 00:02)