https://touchingfish.top/tags/sequences/Recent content in Sequences on TouchingFish.topHugozh-cnWed, 09 Jun 2021 00:00:00 +0000https://touchingfish.top/calculus101/2021-cal7/Wed, 09 Jun 2021 00:00:00 +0000https://touchingfish.top/calculus101/2021-cal7/<h1 id="数列和级数基本概念">§数列和级数：基本概念</h1> <ul> <li>数列（sequences）的收敛和发散</li> <li>两个重要的数列</li> <li>数列极限与函数极限的关系</li> <li>级数的收敛与发散，以及几何级数（geometric series）的敛散性</li> <li>第 $n$ 项判别法（the $n$th term test for series）</li> <li>级数和反常积分的联系</li> <li>比式判别法（ratio test）、根式判别法（root test）、积分判别法（integral test）和交错级数判别法（alternating series test）</li> </ul> <h2 id="221-数列的收敛和发散">22.1 数列的收敛和发散</h2> <p>无穷数列（infinite sequence）的敛散性</p> <blockquote> <p>In math notation, does $\lim\limits_{n\to\infty}a_n$ exist, and if so, what is it?</p> </blockquote> <h3 id="2211-数列和函数的关系">22.1.1 数列和函数的关系</h3> <blockquote> <p>There&rsquo;s also a connection to horizontal asymptotes:</p> </blockquote> <p>若 $\lim\limits_{x\to\infty}f(x)=L$，则 $y=f(x)$ 的图像有水平渐近线 $y=L$</p> <blockquote> <p>For example, if you have two convergent sequences $a_n$ and $b_n$, such that $a_n \to L$ and $b_n \to M$ as $n \to \infty$, then the sum $a_n + b_n$ gives a new sequence which converges to $L + M$. The same goes for differences, products, quotients (provided that $M \neq 0$, since you can&rsquo;t divide by $0$), and constant multiples.</p>