【数学分析】520问题解答

2023-05-21 23:25 作者:Ice_koucha 0人读过 | 我要投稿

昨天(5月20号)的动态Ice_koucha的动态 - 哔哩哔哩 (bilibili.com)

问题： $%5Cfrac%7B%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(2023-x)%7D%20%7D%7B%5Csqrt%7Bln(189%2Bx)%7D%20%2B%5Csqrt%7Bln(2023-x)%7D%7Ddx-3%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bj%3D1%7D%5En%20%5Csqrt%5Bj%5D%7Bn%7D%20%20%7D%7Bn%7D%20%20%20-%5Clim_%7Bx%5Cto0%7D%20%5Cint_%7B0%7D%5E%7Bx%7D%5Cfrac%7Bsint%7D%7B%5Csqrt%7B4%2Bt%5E2%7D%20%5Cint_%7B0%7D%5E%7Bx%7D(%5Csqrt%7Bt%2B1%7D-1%20)dt%20%7D%20%20%20dt%7D%7B%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B(n%2B114514)(n-1919810)%7D%7B%5Csum_%7Bi%3D1%7D%5En%5Csqrt%7Bi%7D%20%5Ccdot%20%5Csum_%7Bk%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7Bk%7D%20%7D%20%20%20%7D%20%20%7D%20$

$Part%E2%85%A0$

$%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(2023-x)%7D%20%7D%7B%5Csqrt%7Bln(189%2Bx)%7D%20%2B%5Csqrt%7Bln(2023-x)%7D%7Ddx$

(引理1)区间再现公式：

$%5Cint_%7Ba%7D%5E%7Bb%7D%20f(x)dx%3D%5Cint_%7Ba%7D%5E%7Bb%7D%20f(a%2Bb-x)dx$

证明：令 $a%2Bb-x%3Dt$ ，则 $x%3Da%2Bb-t$

所以 $%5Cint_%7Ba%7D%5E%7Bb%7D%20f(a%2Bb-x)dx%3D%5Cint_%7Bb%7D%5E%7Ba%7D%20f(t)d(a%2Bb-t)%3D%5Cint_%7Ba%7D%5E%7Bb%7D%20f(t)dt$

因为定积分的值与字母无关，证毕。

$%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(2023-x)%7D%20%7D%7B%5Csqrt%7Bln(189%2Bx)%7D%20%2B%5Csqrt%7Bln(2023-x)%7D%7Ddx%3D%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(188%2Bx)%7D%20%7D%7B%5Csqrt%7Bln(2023-x)%7D%2B%5Csqrt%7Bln(189%2Bx)%7D%20%20%7Ddx%20$

$%3D%5Cfrac%7B1%7D%7B2%7D%20(%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(2023-x)%7D%20%7D%7B%5Csqrt%7Bln(189%2Bx)%7D%20%2B%5Csqrt%7Bln(2023-x)%7D%7Ddx%2B%5Cint_%7B520%7D%5E%7B1314%7D%20%5Cfrac%7B%5Csqrt%7Bln(188%2Bx)%7D%20%7D%7B%5Csqrt%7Bln(2023-x)%7D%2B%5Csqrt%7Bln(189%2Bx)%7D%20%20%7Ddx%20)$

$%3D%5Cfrac%7B1%7D%7B2%7D%20%5Cint_%7B520%7D%5E%7B1314%7D%20dx$

$%3D397$

$Part%E2%85%A1$

$%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bj%3D1%7D%5En%20%5Csqrt%5Bj%5D%7Bn%7D%20%7D%7Bn%7D%20$

$%20%20%20%5Coverset%7BStolz%E5%AE%9A%E7%90%86%7D%7B%3D%7D%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bj%3D1%7D%5En%20%5Csqrt%5Bj%5D%7Bn%7D-%5Csum_%7Bj%3D1%7D%5E%7Bn%2B1%7D%20%5Csqrt%5Bj%5D%7Bn%2B1%7D%7D%7Bn%2B1-n%7D%20$

$%3D2%2B%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Csum_%7Bj%3D2%7D%5En%20(%5Csqrt%5Bj%5D%7Bn%2B1%7D-%5Csqrt%5Bj%5D%7Bn%7D%20)$

(引理2)当 $m%3E1$ 时，有下列不等式成立：

$%5Cfrac%7B1%7D%7Bm%7D%5Ccdot%20%5Cfrac%7B1%7D%7B(n%2B1)%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20%20%3C%5Csqrt%5Bm%5D%7Bn%2B1%7D%20-%5Csqrt%5Bm%5D%7Bn%7D%20%3C%5Cfrac%7B1%7D%7Bm%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7Bn%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20$

证明：令 $f(t)%3D%5Csqrt%5Bm%5D%7Bt%7D%20(t%5Cgeq%201)$ ，在 $t%3Dx%2B1$ 和 $t%3Dx$ 上使用 $Lagrange$ 中值定理，有

$f(x%2B1)-f(x)%3Df'(%20%5Cxi%20)%3D%5Cfrac%7B1%7D%7Bm%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7B%5Cxi%20%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20$ (其中 $%5Cxi%20%5Cin(x%2Cx%2B1)$ )

令 $g(x)%3D%5Cfrac%7B1%7D%7Bx%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%7D%20(x%5Cgeq%201)$ ，显然 $g(x)$ 严格单调递减。

所以有

$%5Cfrac%7B1%7D%7Bm%7D%5Ccdot%20%5Cfrac%7B1%7D%7B(x%2B1)%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20%20%3C%5Cfrac%7B1%7D%7Bm%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7B%5Cxi%20%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20%20%3C%5Cfrac%7B1%7D%7Bm%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7Bx%5E%7B1-%5Cfrac%7B1%7D%7Bm%7D%20%7D%20%7D%20$

将上述不等式中 $x$ 换为 $n$ ，证毕。

$0%3C%5Csum_%7Bj%3D2%7D%5En%20(%5Csqrt%5Bj%5D%7Bn%2B1%7D-%5Csqrt%5Bj%5D%7Bn%7D%20)%3C%5Csum_%7Bj%3D2%7D%5En%20%5Cfrac%7B1%7D%7Bj%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7Bn%5E%7B1-%5Cfrac%7B1%7D%7Bj%7D%20%7D%20%7D%20%3C%5Cfrac%7B%5Csum_%7Bj%3D2%7D%5En%20%5Cfrac%7B1%7D%7Bj%7D%20%7D%7B%5Csqrt%7Bn%7D%20%7D%20$

而 $%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bj%3D2%7D%5En%20%5Cfrac%7B1%7D%7Bj%7D%20%7D%7B%5Csqrt%7Bn%7D%20%7D%5Coverset%7BStolz%E5%AE%9A%E7%90%86%7D%7B%3D%7D%0A%5Clim_%7Bn%5Cto%E2%88%9E%7D%5Cfrac%7B%5Csum_%7Bj%3D2%7D%5E%7Bn%2B1%7D%5Cfrac%7B1%7D%7Bj%7D-%5Csum_%7Bj%3D2%7D%5En%20%5Cfrac%7B1%7D%7Bj%7D%20%7D%7B%5Csqrt%7Bn%2B1%7D-%5Csqrt%7Bn%7D%20%7D%20%3D2%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csqrt%7Bn%7D%20%7D%7Bn%2B1%7D%20%3D0$

由夹逼定理知 $%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Csum_%7Bj%3D2%7D%5En%20(%5Csqrt%5Bj%5D%7Bn%2B1%7D-%5Csqrt%5Bj%5D%7Bn%7D%20)%3D0$

所以 $%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bj%3D1%7D%5En%20%5Csqrt%5Bj%5D%7Bn%7D%20%7D%7Bn%7D%20%3D2$

$Part%E2%85%A2$

$%5Clim_%7Bx%5Cto0%7D%20%5Cint_%7B0%7D%5E%7Bx%7D%5Cfrac%7Bsint%7D%7B%5Csqrt%7B4%2Bt%5E2%7D%20%5Cint_%7B0%7D%5E%7Bx%7D(%5Csqrt%7Bt%2B1%7D-1%20)dt%20%7D%20%20dt$

$%5Coverset%7BL'H%C3%B4pital%7D%7B%3D%7D%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7Bsinx%7D%7B%5Csqrt%7B4%2Bx%5E2%20%7D(%5Csqrt%7Bx%2B1%7D-1%20)%20%7D%20$

$%3D%5Cfrac%7B1%7D%7B2%7D%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7Bx%7D%7B%5Csqrt%7Bx%2B1%7D-1%20%7D%20%20$

$%3D%5Cfrac%7B1%7D%7B2%7D%20%5Clim_%7Bx%5Cto0%7D%20%5Cfrac%7Bx%7D%7B%5Cfrac%7B1%7D%7B2%7Dx%20%7D%20$

$%3D1$

$Part%E2%85%A3$

$%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B(n%2B114514)(n-1919810)%7D%7B%5Csum_%7Bi%3D1%7D%5En%5Csqrt%7Bi%7D%20%5Ccdot%20%5Csum_%7Bk%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7Bk%7D%20%7D%20%20%20%7D%20%20$

$%3D%5Cfrac%7B1%7D%7B%5Clim_%7Bn%5Cto%E2%88%9E%7D%20%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%5Csqrt%7Bi%7D%20%5Ccdot%20%5Csum_%7Bk%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7Bk%7D%20%7D%20%7D%7Bn%5E2%20%7D%20%20%7D%20$

$%3D%5Cfrac%7B1%7D%7B%5Clim_%7Bn%5Cto%E2%88%9E%7D%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%5Csqrt%7Bi%7D%20%20%7D%7Bn%5E%5Cfrac%7B3%7D%7B2%7D%20%20%7D%5Ccdot%20%20%20%5Clim_%7Bn%5Cto%E2%88%9E%7D%5Cfrac%7B%5Csum_%7Bk%3D1%7D%5En%5Cfrac%7B1%7D%7B%5Csqrt%7Bk%7D%20%7D%20%20%7D%7Bn%5E%7B%5Cfrac%7B1%7D%7B2%7D%20%7D%20%7D%20%7D%20$

$%3D%5Cfrac%7B1%7D%7B%5Cint_%7B0%7D%5E%7B1%7D%5Csqrt%7Bx%7D%20dx%5Ccdot%20%20%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%7D%20%7Ddx%20%20%7D%20$