【站僕】摩檸Morning>試卷(2016/08/29)

# 104 年 - 104 淡江大學 轉學考 高等微積分#55495

【非選題】
1.1(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.

【非選題】
2.
2(10%) Let a1± = 1 and an+1 = for n = 1,2,3,...

【題組】(a) Show that an is monotone increasing (by induction).

【非選題】
3.【題組】(b) Show that an is bounded above by 2 (by induction).

【非選題】
4.【題組】(c) Find the limn→∞ an.

【非選題】
5.3(10%)  is open or give a counterexample.

【非選題】
6.4(10%) Test for convergence or divergence

【非選題】
7.5(10%) Classify the following series into absolute convergence, conditional convergence or divergence. Explain why.

【非選題】
8.
6(20%)

【題組】 (a) Give the definition of uniform continuity of a function f defined on a set E .

【非選題】
9.【題組】(b) Give the definition of a set E being “compactness” in .

【非選題】
10.【題組】(c) Show that if f is continuous on a compact set E M, then f is uniformly continuous.

【非選題】
11.
7(10%)

【題組】(a)

【非選題】
12.【題組】(b)

【非選題】
13.【已刪除】【題組】(c)

【非選題】
14.
8. (10%)

【題組】(a) Find fx(0,0) and fy,(0,0).

【非選題】
15.【題組】(b) Find the partial derivative of / at (0,0) with respect to any vector  =(a,b) is

【非選題】
16.【題組】(c) Show that / is not differentiable at (0,0).

【非選題】
17.【已刪除】
9. (10%) Evaluate

【題組】(a) Find the limit function f{x).

【非選題】
18.【已刪除】【題組】(b) Does fn(x) converge to f{x) uniformly? Explain why.

### 懸賞詳解

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