作者rod24574575 (天然呆)
看板NTU-Exam
标题[试题] 99下 薛克民 微积分甲下 第二次小考
时间Sun Apr 3 16:51:38 2011
课程名称︰微积分甲下
课程性质︰必修
课程教师︰薛克民
开课学院：电资学院、工学院、管理学院
开课系所︰电机系、资工系、材料系、资管系
考试日期（年月日）︰100/3/7
考试时限（分钟）：40分钟
是否需发放奖励金：是
（如未明确表示，则不予发放）
试题 :
Write down all the details or you may
not get the full grades.
There could be some problems on the
back of the paper.
x sin(t)
A. (a) Find Taylor series for f(x) = ∫ ──── dt and determine the
0 t
interval of convergence. (5%)
x ∞ (-1)^(n-1)
(b) Show that ∫ x^(x) dx = Σ ────── . Hint: Integrate a series
0 n=1 n^(n)
term by term. (5%)
B. Find the sum of
∞ n ∞ n^(2)
(a) Σ ────(5%) (b) Σ ────(5%)
n=1 2^(n) n=1 2^(n)
∞
Hint: Starting with the geometric series Σ x^(n)
n=0
C. (a) Find the Taylor series for ㏑(1+x) (5%)
(1+x)
(b) Find the Taylor series for e (5%)
(1+x) x
Hint: e = e．e
1 x 1
(c) Find the Taylor series for (1 +
──) in porwers of ──to degree
x x
three. (5%)
Hint: Consider it in exponential form and use the rusults of (a), (b).
╭ e 2┌ 1 x ┐ 11 ╮
(d) Evaluate the limit: lim x│
──*x + x │ (1 +
──) - e │- ──*e │
x→∞ ╰ 2 └ x ┘ 24 ╯
(5%)
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