104年 - 104 淡江大學 轉學考 高等微積分#55495

科目：研究所、轉學考(插大)-高等微積分 | 年份：104年 | 選擇題數：0 | 申論題數：18

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所屬科目：研究所、轉學考(插大)-高等微積分

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申論題 (18)

【已刪除】1(10%) A sequence {xn} in  is said to be a Cauchy sequence in case for every ε ＞ 0, there is a natural number n0 such that for all m,n＞ n0, then |xm — xn  |＜ ε. Show that a convergent sequence is a Cauchy sequence.
(a) Show that an is monotone increasing (by induction).
(b) Show that an is bounded above by 2 (by induction).
(c) Find the limn→∞ an.
【已刪除】3(10%)  is open or give a counterexample.
4(10%) Test for convergence or divergence
5(10%) Classify the following series into absolute convergence, conditional convergence or divergence. Explain why.
(a) Give the definition of uniform continuity of a function f defined on a set E .
(b) Give the definition of a set E being “compactness” in .
(c) Show that if f is continuous on a compact set E M, then f is uniformly continuous.
【已刪除】(a)
(b)
【已刪除】(c)
(a) Find fx(0,0) and fy,(0,0).
【已刪除】(b) Find the partial derivative of / at (0,0) with respect to any vector  =(a,b) is
(c) Show that / is not differentiable at (0,0).
【已刪除】(a) Find the limit function f{x).
【已刪除】(b) Does fn(x) converge to f{x) uniformly? Explain why.