课程名称︰微积分甲下
课程性质︰大一必修
课程教师︰容志辉
开课学院：工学院
开课系所︰电机系、材料系
考试日期（年月日）︰2015/3/23
考试时限（分钟）：50 分钟
试题 :
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CALCULUS II QUIZ 1, 2015 SPRING
(Total 150 points)
1.(20%+20%+20%) Determine whether the following is convergent or divergent
and explain briefly.
2
(a) a_n = √( n + n ) - n
1 2
(b) a_1 = -2 and a_n+1 = ──( a_n + ── ) for all n ∈ N.
2 a_n
∞ 1
(c) Σ ───────
n=3 n √( ㏑ n )
2.(15%+15%+15%+15%) True of False. If it is right, justify your answer. If
it is false, disprove it or give a counterexample.
(a) Given two sequences such that a_n < b_n for all n ∈ N and a_n → a,
b_n → b. Then a < b.
2
(b) If a_n <= 1 / n for all n ∈ N, then Σa_n converges.
(c) If a_n > 0 for all n ∈ N and Σa_n converges, then Σsin(a_n) is also
convergent.
(d) Every Cauchy sequence (a_n) in R is contained in some open interval
(a,b), that is a_n ∈ (a,b) for all n ∈ N.
2 -1/2 2
3.(20%+10%) Suppose that ∫( 1 - x ) dx is expressed as a + bx + cx
3 4 5
+ dx + ex + O(x ). Find the coefficient a, b, c, d, e and its radius of
a a(a-1)…(a-n+1) a
convergence. [Note: ( ) = ───────── if n ≠ 0 and ( ) = 1. ]
n n! 0
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