(AMM. Problems and Solutions. 2015. 03) Let $\sed{a_n}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $$\bex \vsm{n}a_n<\infty \eex$$ if and only if $a_n=O(1/\ln n)$ and $\dps{\vsm{n}(a_n-a_{n+1}) \ln n<\infty}$…

设非负严格增加函数 $f$ 在区间 $[a,b]$ 上连续, 有积分中值定理, 对于每个 $p>0$ 存在唯一的 $x_p\in (a,b)$, 使 $$\bex f^p(x_p)=\cfrac{1}{b-a}\int_a^b f^p(t)\rd t. \eex$$ 试求 $\dps{\vlm{p}x_p}$. 解答: 由 H\"older 不等式, $$\beex \bea f^p(x_p)&=\cfrac{1}{b-a}\int_a^b f^p(t)\cdot 1\rd t\\…

(2014-04-08 from 1297503521@qq.com) 设方程 $\sin x-x\cos x=0$ 在 $(0,+\infty)$ 中的第 $n$ 个解为 $x_n$. 证明: $$\bex n\pi+\cfrac{\pi}{2}-\cfrac{1}{n\pi} <x_n<n\pi+\cfrac{\pi}{2}. \eex$$ 证明: 设 $f(x)=\sin x-x\cos x$, 则 $$\bex f'(x)=x\sin x\sedd{\ba{ll} >0,&…

试证: $$\bex \left(1+\frac{1}{x}\right)^x>\frac{2ex}{2x+1},\forall\ x>0. \eex$$ 证明 (from Hansschwarzkopf): 对任何$x>0$, 有 \[x\ln\left(1+\frac{1}{x}\right)=x\ln\frac{1+\frac{1}{2x+1}}{1-\frac{1}{2x+1}} =2x\left(\frac{1}{2x+1}+\frac{1}{3(2x+1)^3}+\ldots…

设 $A,B\in \bbR^{n\times n}$ 的特征值都是实数, 则存在正交阵 $P,Q$ 使得 $PAQ$, $PBQ$ 为上三角阵.…

设幂级数 $\dps{g(x)=\sum_{n=0}^\infty a_nx^n}$ 在 $|x|<1$ 内收敛, 且 $\dps{\sum_{n=0}^\infty a_n=s}$ 收敛. 则 $$\bex \lim_{x\to 1^-} g(x)=s. \eex$$ 证明: 记 $s_n=a_0+\cdots +a_n$, 则 $\dps{\vlm{n}s_n=s}$. 写出 $$\beex \bea \sum_{k=0}^n a_kx^k &=a_0+\sum_{k=1}^n (s_k…

$$\bex \sin(x+y)=\sin x\cos y+\cos x\sin y. \eex$$ Ref. [Proof Without Words: Sine Sum Identity, The College Mathematics Journal].…

$$\bex \frac{\sin x}{x}\nearrow. \eex$$ Ref. [Proof Without Words: Monotonicity of $\sin x/x$ on $(0,\pi/2)$, The College Mathematics Journal]…

$$\bex \frac{\tan x}{x}\nearrow. \eex$$ Ref. [Proof Without Words: Monotonicity of $\tan x/x$ on $(0,\pi/2)$, The College Mathematics Journal].…

设 $f$ 是 $\bbR$ 上周期为 $1$ 的连续可微函数, 满足 $$\bee\label{141102_f} f(x)+f\sex{x+\frac{1}{2}}=f(2x),\quad\forall\ x. \eee$$ 试证: $f(x)=0$, $\forall\ x$. 证明: (from xida that this proof comes from ``Proofs of the book'' 4th edition, Chapter 23) 设 $g(x)=f'(x)$, 则…