Infinity

求极限 limn→∞1n⎛⎝n12+23+⋯+nn+1⎞⎠n. \lim_{n\to \infty} \frac{1}{n} \left( \frac{n}{\frac{1}{2}+\frac{2}{3}+\cdots+\frac{n}{n+1} }\right)^n. 解. 利用 1+12+13+⋯+1n=lnn+γ+o(1),n→∞.1+\frac{1}{2}+\frac{1}{3}+\c

求极限 limn→∞2n2−2+2+⋯2√−−−−−−−−√−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−√.\lim_{n\to \infty} 2^n \sqrt{2-\sqrt{2+\sqrt{2+\cdots\sqrt 2}}} . 解. 2√=2cosπ4=2cosπ222+2√−−−−−−√=2(1+cosπ22)−−−−−−−−−−−√=2cosπ23⋯2−2+

B(x,y)=Γ(x)Γ(y)Γ(x+y).\mathrm{B}(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. 其中 Γ(x)=∫+∞0e−ttx−1dt,B(x,y)=∫10tx−1(1−t)y−1dt.\Gamma(x)=\int_0^{+\infty} e^{-t} t^{x-1} \mathrm{d}t,\quad B(x,y)=\int_0^

设 f(x)f(x) 是闭区间 [0,1][0,1] 上满足 f(0)=f(1)=0f(0)=f(1)=0 的连续可微函数，求证不等式 (∫10f(x)dx)2≤112∫10|f′(x)|2dx,\left(\int_0^1 f(x)\mathrm{d}x \right)^2 \leq \frac{1}{12} \int_0^1 |f'(x)|^2 \mathrm{d}x, 并且等号成立当且仅当