https://cn-sunisalex-pages.zh-cn.edgeone.cool/categories/%E9%AB%98%E8%80%83%E6%95%B0%E5%AD%A6/Recent content in 高考数学 on ∇Alex的实验室Hugozh-CNFri, 03 Apr 2026 20:48:40 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/2024-fujian-xiamen-sanmo-derivative/Fri, 03 Apr 2026 20:48:40 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/2024-fujian-xiamen-sanmo-derivative/本文围绕对数函数 <span class="math math-inline"> $\ln x$ </span> 与 <span class="math math-inline"> $\ln(1+x)$ </span> 的 <strong>Padé 逼近与不等式放缩</strong> 展开，系统总结了一类高考常见的估值与比较技巧。https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/2026-3-27-lwy-review/Fri, 27 Mar 2026 22:17:31 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/2026-3-27-lwy-review/<h1 id="两个重要数列">两个重要数列</h1> <p><span class="math math-inline"> $n\in N^*$ </span>,有以下两个数列:</p> <ul> <li><span class="math math-inline"> $a_n=(1+\frac{1}{n})^n$ </span></li> <li><span class="math math-inline"> $b_n=(1+\frac{1}{n})^{n+1}$ </span></li> </ul> <p>我们有以下两个结论:</p> <ol> <li><span class="math math-inline"> $a_n\lt e\lt b_n$ </span></li> <li><span class="math math-inline"> $\{a_n\}$ </span>为<strong>递增</strong>数列,<span class="math math-inline"> $\{b_n\}$ </span>为<strong>递减</strong>数列</li> <li><span class="math math-inline"> $\{a_n\},\{b_n\}$ </span><strong>收敛</strong>,<span class="math math-inline"> $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=e$ </span></li> </ol> <p>我们可以观看<strong>优美的desmos图像</strong>感受性质. <div class="desmos-container" id="desmos-calc-1778194069773071238" data-desmos-id="calc-1778194069773071238" data-funcs="a_{n}=\left(1&#43;\frac{1}{n}\right)^{n}\left\{n\ge1\right\}|b_{n}=\left(1&#43;\frac{1}{n}\right)^{n&#43;1}\left\{n\ge1\right\}|y=e\left\{n\ge1\right\}" data-xmin="0" data-xmax="10" data-ymin="0" data-ymax="4" style="width:100%; height:500px; border:2px solid #ccc;"> </div> </p>https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/lnx-approximation/Thu, 05 Mar 2026 20:54:13 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/lnx-approximation/<h1 id="泰勒展开">泰勒展开</h1> <p><span class="math math-inline"> $ \ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^{n-1} \frac{x^n}{n} + o(x^n) $ </span> 需要注意的是,此公式具有收敛半径<span class="math math-inline"> $R=1$ </span>,仅适用于<span class="math math-inline"> $x\approx 0$ </span>的情况.</p> <p>具体内容,可以见<a href="https://www.sunisalex.org/posts/math/taylor-expansion/">浅谈泰勒展开与高考数学</a>.</p> <h1 id="帕德逼近">帕德逼近(※)</h1> <p>以下是<a href="https://zh.wikipedia.org/zh-sg/%E5%B8%95%E5%BE%B7%E8%BF%91%E4%BC%BC">维基百科</a>给出的简介,这里不加赘述. <figure class="img-wrapper"> <a href="https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/lnx-approximation/AgACAgEAAyEGAATlvtEQAAMeaal9j4ifvzXpCZbNC-5wPK-cSREAAukLaxuKrkhFJmRRegg0MOEBAAMCAAN3AAM6BA.png" target="_blank" rel="noopener"> <div class="loading-spinner"></div> <img alt="图片" loading="lazy" src="https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/lnx-approximation/AgACAgEAAyEGAATlvtEQAAMeaal9j4ifvzXpCZbNC-5wPK-cSREAAukLaxuKrkhFJmRRegg0MOEBAAMCAAN3AAM6BA.png"style="opacity: 0; transition: opacity 0.4s ease;" onload="this.style.opacity='1'; if (this.previousElementSibling &amp;&amp; this.previousElementSibling.classList.contains('loading-spinner')) { this.previousElementSibling.style.display='none'; }"> </a></figure> </p>https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/taylor-expansion/Mon, 23 Feb 2026 17:10:57 +0800https://cn-sunisalex-pages.zh-cn.edgeone.cool/posts/math/taylor-expansion/<h1 id="泰勒公式">泰勒公式</h1> <h2 id="高数知识">高数知识</h2> <p>泰勒 (Taylor) 公式的主要作用是用多项式逼近函数和近似计算，对应的分别是带有皮亚诺余项的泰勒公式和带有拉格朗日余项的泰勒公式。</p> <h3 id="1-带有皮亚诺余项的泰勒公式">1. 带有皮亚诺余项的泰勒公式</h3> <p>若函数 <span class="math math-inline"> $ f(x) $ </span> 在点 <span class="math math-inline"> $ x_0 $ </span> 处存在直至 <span class="math math-inline"> $ n $ </span> 阶导数，则有 </p>