复习笔记Day3

2022-08-24 04:28 作者:臥煙鈴 0人读过 | 我要投稿

今天也和之前一样给带火分享一道高代和一道数分

$9.(%E4%B8%AD%E5%B1%B1%E5%A4%A7%E5%AD%A62017)$

$%E8%AE%BEA%E6%98%AF%E4%B8%80%E4%B8%AAn%E9%98%B6%E7%9A%84%E5%AE%9E%E7%9F%A9%E9%98%B5%EF%BC%8C%E6%9C%89n%E4%B8%AA%E7%BB%9D%E5%AF%B9%E5%80%BC%E5%B0%8F%E4%BA%8E1%E7%9A%84%E5%AE%9E%E7%89%B9%E5%BE%81%E5%80%BC$ $%E8%AF%81%E6%98%8E%EF%BC%9Aln(det(I-A))%3D%5Csum_%7Bk%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B1%7D%7Bk%7Dtr(A%5Ek)$

$%E8%AF%81%EF%BC%9A$

$%E8%AE%BEA%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC%E5%88%86%E5%88%AB%E4%B8%BA%5Clambda_1%2C%5Cldots%2C%5Clambda_n$

$%E6%88%91%E4%BB%AC%E5%AE%B9%E6%98%93%E5%BE%97%E5%88%B0%7C%5Clambda_jI-A%7C%3D0%2Cj%3D1%2C2%2C%5Cldots%2Cn$ $%E8%80%8C%E7%94%B1%7C%5Clambda%20I-A%7C(%7C%5Clambda%5E%7Bn-1%7DI%2B%5Clambda%5E%7Bn-2%7DA%2B%5Ccdots%2B%5Clambda%20A%5E%7Bn-2%7D%2BA%5E%7Bn-1%7D%7C)$

$%3D%7C(%5Clambda%20I-A)(%5Clambda%5E%7Bn-1%7DI%2B%5Clambda%5E%7Bn-2%7DA%2B%5Ccdots%2B%5Clambda%20A%5E%7Bn-2%7D%2BA%5E%7Bn-1%7D)%7C%3D%7C%5Clambda%5EnI-A%5En%7C$

$%E4%BA%8E%E6%98%AF%7C%5Clambda_j%5EkI-A%5Ek%7C%3D%7C(%5Clambda_j%20I-A)(%5Clambda_j%5E%7Bk-1%7DI%2B%5Clambda_j%5E%7Bk-2%7DA%2B%5Ccdots%2B%5Clambda_j%20A%5E%7Bk-2%7D%2BA%5E%7Bk-1%7D)%7C%3D0$

$%E6%89%80%E4%BB%A5%5Clambda_j%5Ek%E6%98%AFA%5Ek%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC(j%3D1%2C2%2C%5Cldots%2Cn)$

$%E4%BA%8E%E6%98%AF%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B1%7D%7Bk%7Dtr(A)%5Ek%3D%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B1%7D%7Bk%7D(%5Clambda_1%5Ek%2B%5Clambda_2%5Ek%2B%5Ccdots%2B%5Clambda_n%5Ek)$

$%E7%94%B1%E4%BA%8E%E7%89%B9%E5%BE%81%E5%80%BC%E9%83%BD%E5%B0%8F%E4%BA%8E1%EF%BC%8C%E6%89%80%E4%BB%A5%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B1%7D%7Bk%7D(%5Clambda_1%5Ek%2B%5Clambda_2%5Ek%2B%5Ccdots%2B%5Clambda_n%5Ek)$ $%3Dln(1-%5Clambda_1)%2Bln(1-%5Clambda_2)%2B%5Ccdots%2Bln(1-%5Clambda_n)%3Dln%5B%5Cprod_%7Bi%3D1%7D%5En%20(1-%5Clambda_i)%5D%0A$

$%E6%88%91%E4%BB%AC%E5%8F%88%E8%83%BD%E5%A4%9F%E5%AE%B9%E6%98%93%E7%9F%A5%E9%81%93I-A%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC%E5%88%86%E5%88%AB%E6%98%AF1-%5Clambda_1%2C%5Cldots%EF%BC%8C1-%5Clambda_n$

$%E4%BA%8E%E6%98%AF%5Csum_%7Bk%3D1%7D%5E%5Cinfty%5Cfrac%7B1%7D%7Bk%7Dtr(A)%5Ek%3Dln%5B%5Cprod_%7Bi%3D1%7D%5En%20(1-%5Clambda_i)%5D%3Dln(det(I-A))$

$%5Cmathbb%7BQ.E.D%7D$

这道题也就看上去吓人了那么点而已，只不过是稍微把高代和级数的知识缝合了一下罢了

接下来是一道相当传统的数列极限题

$1.(%E4%B8%AD%E5%9B%BD%E4%BA%BA%E6%B0%91%E5%A4%A7%E5%AD%A62022)$

$%E5%B7%B2%E7%9F%A5f_n(x)%3Dx%2Bx%5E2%2B%5Ccdots%2Bx%5En%2C(n%3D2%2C3%2C%5Cldots)$

$%E8%AF%81%E6%98%8E%3A$

$i)%5C%20f_n(x)%3D1%E5%9C%A8(%5Cfrac%7B1%7D%7B2%7D%2C1)%E5%86%85%E6%9C%89%E5%94%AF%E4%B8%80%E5%AE%9E%E6%95%B0%E6%A0%B9a_n%5C%20%3B$

$ii)%5C%20a_n%E6%94%B6%E6%95%9B%EF%BC%8C%E5%B9%B6%E6%B1%82%E5%87%BA%5Clim_%7Bn%5Cto%20%5Cinfty%7Da_n$

$%E8%AF%81%3A$

$i)%5C%20%E4%BB%A4g_n(x)%3Df_n(x)-1$

$g_n(%5Cfrac%7B1%7D%7B2%7D)%3Df_n(%5Cfrac%7B1%7D%7B2%7D)-1%3D%5Cfrac%7B1%7D%7B2%7D%2B(%5Cfrac%7B1%7D%7B2%7D)%5E2%2B(%5Cfrac%7B1%7D%7B2%7D)%5E3%2B%5Ccdots%2B(%5Cfrac%7B1%7D%7B2%7D)%5En-1$

$f_n(%5Cfrac%7B1%7D%7B2%7D)-f_%7Bn-1%7D(%5Cfrac%7B1%7D%7B2%7D)%3D(%5Cfrac%7B1%7D%7B2%7D)%5En%3E0%2Cn%3D3%2C4%2C%5Cldots%2C%E6%95%85f_n(%5Cfrac%7B1%7D%7B2%7D)%E6%98%AF%E9%80%92%E5%A2%9E%E6%95%B0%E5%88%97$

$%5Clim_%7Bn%5Cto%5Cinfty%7Df_n(%5Cfrac%7B1%7D%7B2%7D)%3D%5Cfrac%7B1%7D%7B2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%3D1$

$f_n(%5Cfrac%7B1%7D%7B2%7D)%3C1%2Cn%3D2%2C3%2C%5Cldots%2C%E6%95%85g_n(%5Cfrac%7B1%7D%7B2%7D)%3C0$

$%E8%80%8C%E6%88%91%E4%BB%AC%E5%AE%B9%E6%98%93%E7%9F%A5%E9%81%93f_n(1)%3Dn%E4%B8%94n%3D2%2C3%2C%5Ccdots%0A$

$%E6%95%85f_n(1)%3E1%5Cimplies%20g_n(1)%3E0$

$%E8%80%8C%5Cforall%20x%5Cin(%5Cfrac%7B1%7D%7B2%7D%2C1)%E6%9C%89g'_n(x)%3Df'_n(x)%3D1%2B2x%2B3x%5E2%2B%5Ccdots%2Bnx%5E%7Bn-1%7D%3E0$

$%E6%95%85x%5Cin(%5Cfrac%7B1%7D%7B2%7D%2C1)%E6%9C%89g_n(x)%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E$

$%E7%94%B1g(%5Cfrac%7B1%7D%7B2%7D)%3C0%2Cg(1)%3E0%E6%88%91%E4%BB%AC%E4%BE%BF%E5%BE%97%E5%88%B0%5Cexists%201%20a_n%5Cin(%5Cfrac%7B1%7D%7B2%7D%2C1)st%5C%20g_n(a_n)%3D0$

$ii)%E9%A6%96%E5%85%88%E6%88%91%E4%BB%AC%E7%9F%A5%E9%81%93%5Cfrac%7B1%7D%7B2%7D%3Ca_n%3C1$

$%E4%B8%94%E5%9C%A8(%5Cfrac%7B1%7D%7B2%7D%2C1)%E5%86%85%E6%88%91%E4%BB%AC%E6%9C%89f_n(x)-f_%7Bn-1%7D(x)%3Dx%5En%3E0$

$%E6%AD%A4%E6%97%B6f_n(x)%E6%98%AF%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E5%88%97$

$%E6%88%91%E4%BB%AC%E7%9F%A5%E9%81%93f_n(a_n)%3D1%2C%E6%89%80%E4%BB%A5f_%7Bn%2B1%7D(a_n)%3Ef_n(a_n)%3D1$

$%E7%94%B1f_%7Bn%2B1%7D(a_n)%3E1%3Df_%7Bn%2B1%7D(a_%7Bn%2B1%7D)%E4%B8%94f_n(x)%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E6%88%91%E4%BB%AC%E5%BE%97%E7%9F%A5$

$a_n%3Ea_%7Bn%2B1%7D%EF%BC%8C%E6%95%85a_n%E5%8D%95%E8%B0%83%E9%80%92%E5%87%8F%EF%BC%8C%E8%80%8C%E5%8F%88%E5%9B%A0%E4%B8%BAa_n%E6%9C%89%E7%95%8C%EF%BC%8C%E6%89%80%E4%BB%A5a_n%E6%94%B6%E6%95%9B$