課程名稱︰分析導論優一
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/12/02
考試時限（分鐘）：180
試題 :
1. (20%) Let f(t) be a function defined on [0, 1] with f(t) = (sin t)/t for t >
1
0. Show that I = ∫f(t)dt exists. Without using Taylor's series of sin t,
0
derive that
3
I < ─- cos 1
2
1 sin xt
You need to justify all your arguments. Hint: consider g(x) = ∫ ─── dt.
0 t
2. (20%) Show that for every x ∈ (0, 2π), the series
inx
∞ e
Σ ──
n=1 n
converges and conclude that both series
∞ sin nx ∞ cos nx
Σ ─── and Σ ───
n=1 n n=1 n
n ikx
converge for each x ∈ (0, 2π). Hint: you may need to estimate Σ e .
k=1
3. (20%) Show by an example that a continuous function on [a, b] is not
necessarily of bounded variation on [a, b]. How about if we replace
"continuity" by "differentiability"?
∞ ∞
4. (a) (10%) If Σ a diverges, must Σ (log n) a diverge too?
n=2 n n=2 n
∞
(b) (10%) Given that Σ a converges, where each a > 0. Prove that
n=1 n n
∞ 1/2
Σ (a a )
n=1 n n+1
also converges. Show that the converse is also true if {a } is monotonic.
n
5. (20%) Let f and g be continuous functions mapping from [0, 1] to itself.
Assume that the compositions of f and g are commutative, i.e., f。g = g。f.
Then f and g agree at some point of [0, 1]. Hint: proved by contradiction.