101年 - 101 國立高雄大學_博士班招生考試試題_應用數學系(一般生、在職生)：高等微積分#147990

(1) (15 %) Let $f : [a, b] \to \mathbb{R}$ be continuous and differentiable on $(a, b)$. Assume that $f(a) = 0$ and $\int_{a}^{b} f(x) \, dx = 0$. Prove that there is a $c \in (a, b)$ such that $f'(c) = 0$.

(2) (15 %) Let $\{fn\}$ be a sequence of continuous functions which converge uniformly to a function $f$ on a set $E$. Prove that $\lim{n \to \infty} fn(xn) = f(x)$ for every sequence of points $xn \in E$ such that $xn \to x$ and $x \in E$.

(3) (15 %) Let $M$ be the normed space of all real-valued continuous functions on $[0, 1]$ with the supremum norm. Let $B \subset M$ such that for any $f \in M$, $f$ is $C^1$ on $[0, 1]$, $f(0) = 0$, and $|f'(x)| \leq 1$. Show that the closure of $B$ in $M$ is compact.

(a) (10%) Prove that a real-valued continuous function $f$ on $[a, b]$ is Riemann integrable.

(b) (10%) Prove that an increasing function $f$ on $[a, b]$ is Riemann integrable.

(5) (10 %) If $f$ is continuous on $[0, 1]$ and if $\int_{0}^{1} f(x) x^n \, dx = 0$ ($n = 0, 1, 2, \ldots$), prove that $f(x) = 0$ on $[0, 1]$.

(a) (15%) Prove that $\liminf{n \to \infty} an \leq \liminf{n \to \infty} \sigman \leq \limsup{n \to \infty} \sigman \leq \limsup{n \to \infty} an$.

(b) (10 %) Let $a_n = n^n/n!$. Show that

$$\lim_{n \to \infty} \frac{n}{(n!)^{1/n}} = e.$$