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. -- 第01話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了 第03話 已經沒什麼好期望了 第04話 被當、21都是存在的 第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊 第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋 第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題 第11話 最後留下的補考 第12話 我最愛的學分 --

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