<!doctype html>
<html class="no-js" lang="en">
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<title>
两个数列不等式问题 - 语时lab
</title>
<link href="atom.xml" rel="alternate" title="语时lab" type="application/atom+xml">
<link rel="stylesheet" href="asset/css/foundation.min.css" />
<link rel="stylesheet" href="asset/css/docs.css" />
<script src="asset/js/vendor/modernizr.js"></script>
<script src="asset/js/vendor/jquery.js"></script>
<script src="asset/highlightjs/highlight.pack.js"></script>
<link href="asset/highlightjs/styles/github.css" media="screen, projection" rel="stylesheet" type="text/css">
<script>hljs.initHighlightingOnLoad();</script>
<script type="text/javascript">
function before_search(){
var searchVal = 'site:cshishaliu.github.io ' + document.getElementById('search_input').value;
document.getElementById('search_q').value = searchVal;
return true;
}
</script>
</head>
<body class="antialiased hide-extras">
<div class="marketing off-canvas-wrap" data-offcanvas>
<div class="inner-wrap">
<nav class="top-bar docs-bar hide-for-small" data-topbar>
<section class="top-bar-section">
<div class="row">
<div style="position: relative;width:100%;"><div style="position: absolute; width:100%;">
<ul id="main-menu" class="left">
<li id=""><a target="_self" href="index.html">Home</a></li>
<li id=""><a target="_self" href="book.html">Books</a></li>
<li id=""><a target="_self" href="links.html">Links</a></li>
<li id=""><a target="_self" href="archives.html">Archives</a></li>
<li id=""><a target="_self" href="about.html">About</a></li>
<li id=""><a target="_self" href="todo.html">Todo</a></li>
</ul>
<ul class="right" id="search-wrap">
<li>
<form target="_blank" onsubmit="return before_search();" action="https://google.com/search" method="get">
<input type="hidden" id="search_q" name="q" value="" />
<input tabindex="1" type="search" id="search_input" placeholder="Search"/>
</form>
</li>
</ul>
</div></div>
</div>
</section>
</nav>
<nav class="tab-bar show-for-small">
<a href="javascript:void(0)" class="left-off-canvas-toggle menu-icon">
<span> 语时lab</span>
</a>
</nav>
<aside class="left-off-canvas-menu">
<ul class="off-canvas-list">
<li><a target="_self" href="index.html">Home</a></li>
<li><a target="_self" href="book.html">Books</a></li>
<li><a target="_self" href="links.html">Links</a></li>
<li><a target="_self" href="archives.html">Archives</a></li>
<li><a target="_self" href="about.html">About</a></li>
<li><a target="_self" href="todo.html">Todo</a></li>
<li><label>Categories</label></li>
<li><a href="problemsolving.html">解题</a></li>
<li><a href="mathpost.html">数学随笔</a></li>
<li><a href="Dev.html">Dev</a></li>
<li><a href="Games.html">Games</a></li>
<li><a href="obsolete.html">obsolete</a></li>
<li><a href="misc.html">misc</a></li>
</ul>
</aside>
<a class="exit-off-canvas" href="#"></a>
<section id="main-content" role="main" class="scroll-container">
<script type="text/javascript">
$(function(){
$('#menu_item_index').addClass('is_active');
});
</script>
<div class="row">
<div class="large-8 medium-8 columns">
<div class="markdown-body article-wrap">
<div class="article">
<h1>两个数列不等式问题</h1>
<div class="read-more clearfix">
<span class="date">2019/02/01</span>
<span>posted in </span>
<span class="posted-in"><a href='problemsolving.html'>解题</a></span>
<span class="comments">
</span>
</div>
</div><!-- article -->
<div class="article-content">
<ol>
<li>已知实数 \(x_1, x_2, \dots, x_{10}\), \(\sum\limits_{k=1}^{10} k x_k = 1\). 求 \(\left( \sum\limits_{k=1}^{10} x_k \right)^2 + \sum\limits_{k=1}^{10} x_k^2\) 的最小值.</li>
<li>设 \(a_n = 1 + \dfrac12 + \dfrac13 + \dots + \dfrac1n\). 求证: 对 \(n\ge 2\) 有 \(a_n^2 < 2 \left( \dfrac{a_2}2 + \dfrac{a_3}3 + \dots \dfrac{a_n}n \right) + \dfrac{n+3}{2n+2}\).</li>
</ol>
<span id="more"></span><!-- more -->
<blockquote>
<p>Qer: xieshunfu 20190201</p>
</blockquote>
<h2 id="toc_0">解答</h2>
<h3 id="toc_1">题 1</h3>
<p>对于任意实数 \(t\), 均有 (Cauchy 不等式):<br/>
\[<br/>
\begin{aligned}<br/>
\left[ \left( \sum_{k=1}^{10} x_k \right)^2 + \sum_{k=1}^{10} x_k^2 \right]<br/>
\left[ t^2 + \sum_{k=1}^{10} (k-t)^2 \right] <br/>
&\ge<br/>
\left[ \left( \sum_{k=1}^{10} x_k \right) \cdot t + \sum_{k=1}^{10} (k-t) x_k \right]^2 \\<br/>
&= \left( \sum_{k=1}^{10} k x_k \right)^2 = 1<br/>
\end{aligned}<br/>
\]<br/>
即<br/>
\[<br/>
\left[ \left( \sum_{k=1}^{10} x_k \right)^2 + \sum_{k=1}^{10} x_k^2 \right]<br/>
\cdot (11 t^2 - 110 t + 385) \ge 1<br/>
\]<br/>
或<br/>
\[<br/>
\left( \sum_{k=1}^{10} x_k \right)^2 + \sum_{k=1}^{10} x_k^2 <br/>
\ge \dfrac1{11 t^2 - 110 t + 385}<br/>
\]<br/>
此不等式对一切 \(t \in \mathbb R\) 都成立. 因此<br/>
\[<br/>
\left( \sum_{k=1}^{10} x_k \right)^2 + \sum_{k=1}^{10} x_k^2 <br/>
\ge \left( \dfrac1{11 t^2 - 110 t + 385} \right)_{\max} = <br/>
\left( \dfrac1{11 t^2 - 110 t + 385} \right)_{t=5} = \dfrac1{110}<br/>
\]<br/>
当且仅当 <br/>
\[<br/>
\begin{aligned}<br/>
\left( \sum_{k=1}^{10} x_k \right) : x_1 : x_2 : \dots : x_{10} <br/>
&= t : (1-t) : (2-t) : \dots : (10-t) \\ <br/>
&= 5 : (-4) : (-3) : \dots : 5<br/>
\end{aligned}<br/>
\]<br/>
时取等号.</p>
<h3 id="toc_2">题 2</h3>
<blockquote>
<p>数学归纳法, 不难!</p>
</blockquote>
</div>
<div class="row">
<div class="large-6 columns">
<p class="text-left" style="padding:15px 0px;">
<a href="15840603466913.html"
title="Previous Post: 二元有理函数最值">« 二元有理函数最值</a>
</p>
</div>
<div class="large-6 columns">
<p class="text-right" style="padding:15px 0px;">
<a href="15421980305249.html"
title="Next Post: CMO 2018 题 3">CMO 2018 题 3 »</a>
</p>
</div>
</div>
<div class="comments-wrap">
<div class="share-comments">
</div>
</div>
</div><!-- article-wrap -->
</div><!-- large 8 -->
<div class="large-4 medium-4 columns">
<div class="hide-for-small">
<div id="sidebar" class="sidebar">
<div id="site-info" class="site-info">
<div class="site-a-logo"><img src="https://i.loli.net/2020/02/26/hjpG5rStAgRYm9P.jpg" /></div>
<h1>语时lab</h1>
<div class="site-des">Gnoloac 发文的地方</div>
<div class="social">
<a target="_blank" class="weibo" href="https://weibo.com/gnoloac" title="weibo">Weibo</a>
<a target="_blank" class="github" target="_blank" href="https://github.com/cshishaliu" title="GitHub">GitHub</a>
<a target="_blank" class="email" href="mailto:cshishaliu@163.com" title="Email">Email</a>
<a target="_blank" class="rss" href="atom.xml" title="RSS">RSS</a>
</div>
</div>
<div id="site-categories" class="side-item ">
<div class="side-header">
<h2>Categories</h2>
</div>
<div class="side-content">
<p class="cat-list">
<a href="problemsolving.html"><strong>解题</strong></a>
<a href="mathpost.html"><strong>数学随笔</strong></a>
<a href="Dev.html"><strong>Dev</strong></a>
<a href="Games.html"><strong>Games</strong></a>
<a href="obsolete.html"><strong>obsolete</strong></a>
<a href="misc.html"><strong>misc</strong></a>
</p>
</div>
</div>
<div id="site-categories" class="side-item">
<div class="side-header">
<h2>Recent Posts</h2>
</div>
<div class="side-content">
<ul class="posts-list">
<li class="post">
<a href="15864160277643.html">平面几何的全等和相似符号到底该怎么写</a>
</li>
<li class="post">
<a href="15863333927200.html">数列不等式 `lhc@jinan|20200407`</a>
</li>
<li class="post">
<a href="15863166036092.html">如何判断复合根式是否可以进一步化简</a>
</li>
<li class="post">
<a href="15859891289445.html">Simon Tatham's Portable Puzzle Collection</a>
</li>
<li class="post">
<a href="book.html">Books</a>
</li>
</ul>
</div>
</div>
</div><!-- sidebar -->
</div><!-- hide for small -->
</div><!-- large 4 -->
</div><!-- row -->
<div class="page-bottom clearfix">
<div class="row">
<p class="copyright">Copyright © 2015
Powered by <a target="_blank" href="http://www.mweb.im">MWeb</a>,
Theme used <a target="_blank" href="http://github.com">GitHub CSS</a>.</p>
</div>
</div>
</section>
</div>
</div>
<script src="asset/js/foundation.min.js"></script>
<script>
$(document).foundation();
function fixSidebarHeight(){
var w1 = $('.markdown-body').height();
var w2 = $('#sidebar').height();
if (w1 > w2) { $('#sidebar').height(w1); };
}
$(function(){
fixSidebarHeight();
})
$(window).load(function(){
fixSidebarHeight();
});
</script>
<script src="asset/chart/all-min.js"></script><script type="text/javascript">$(function(){ var mwebii=0; var mwebChartEleId = 'mweb-chart-ele-'; $('pre>code').each(function(){ mwebii++; var eleiid = mwebChartEleId+mwebii; if($(this).hasClass('language-sequence')){ var ele = $(this).addClass('nohighlight').parent(); $('<div id="'+eleiid+'"></div>').insertAfter(ele); ele.hide(); var diagram = Diagram.parse($(this).text()); diagram.drawSVG(eleiid,{theme: 'simple'}); }else if($(this).hasClass('language-flow')){ var ele = $(this).addClass('nohighlight').parent(); $('<div id="'+eleiid+'"></div>').insertAfter(ele); ele.hide(); var diagram = flowchart.parse($(this).text()); diagram.drawSVG(eleiid); } });});</script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-AMS_SVG-full"></script><script type="text/x-mathjax-config">MathJax.Hub.Config({TeX: { equationNumbers: { autoNumber: "AMS" } }});</script>
</body>
</html>