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）︰2014/1/7
考試時限（分鐘）：180
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 105 points.
(1) (15 pts) Determine whether the series is absolutely convergent,
conditionally convergent, or divergent.
n
∞ (-1)
(a) Σ ────
n＝2 2
(log n)
2
∞ n + 2 n
(b) Σ ( ─── )
n＝1 2
3n + 1
∞ π
(c) Σ sin ─
n＝1 ｎ
(2) (10 pts) Evaluate the following limits:
sin x - x
(a) lim ─────
x→0 3
x
x
(b) lim x
x→0+
2
(3) (10 pts) Consider f(x) = x in -π＜x＜π. Determine its Fourier series.
sint
╭ ── , t≠0 x
(4) (10 pts) Consider f(t) = │ ｔ . Let Si(x) =∫f(t)dt. Estimate
╰ 1, t＝0 0
π
Si(─) by using Simpson's Rule with n = 4 and estimate the error.
２
(5) (10 pts) Consider the cycloid given by x(t) = r(t - sin t) and y(t) = r(1 -
cos t). Find its arc length for 0≦t≦2π.
１
(6) (20 pts) Find the Taylor series of f(x) = ─── at x = 0. Determine its
____
√1-x
1
radius of convergence. Also find the Taylor series of f(x) at x = - and
2
determine its radius of convergence.
(7) (10 pts) Let θ, θ, θ be the angles of a triangle. Prove that
1 2 3 _
3√3
sinθ + sinθ + sinθ ≦ ── .
1 2 3 ２
(8) (20 pts) Suppose that f (x) converges to f(x) in the interval [a,b].
n
(a) Give an example that f (x) is continuous in [a,b] for all n but f(x) is
n
not continuous.
(b) If f (x) converges to f(x) in the interval [a,b] uniformly, prove that
n
f (x) is continuous in [a,b] for all n implies that f(x) is continuous.
n
(c) Suppose that f (x) is continuous in [a,b] for all n and converges to
n
b 2 b 2
f(x) uniformly, then ∫f (x) dx converges to ∫f(x) dx.
a n a
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