热门推荐：
声明：本文由入驻搜狐公众平台的作者撰写，除搜狐官方账号外，观点仅代表作者本人，不代表搜狐立场。
　　关注微信：DuoDaaMath每天获得更多数学趣文
　　让我们从一个故事说起：
　　在芝诺(Zeno)五岁那年, 他父亲问他：“从我们家到外婆家一共有五公里的路要走, 如果你以每小时五公里速度去外婆家, 那你需要多长时间才能到？”芝诺回答道：“一小时.”
　　十年之后，芝诺十五岁, 他父亲又问了他一遍相同的问题. 你可能会说, 芝诺父亲怎么会这么无聊？但是芝诺却回答：“永远也走不到.”
　　芝诺是这样回答他父亲的：“如果将五公里的路程一分为二, 那如果想去外婆家就要先走过这段路程的前一半；再将剩下的一半路程一分为二, 如果想要到外婆家, 同样也要走剩下路程的前一半. 之后再将剩下的路程一分为二, 这样无论走久都到不了外婆家, 所以永远也走不到.”
　　这其实就是著名的芝诺悖论, 这个悖论和古希腊神话中的善跑英雄阿喀琉斯(Achilles)追乌龟的故事有着异曲同工之妙. 不过我们现在已经知道这个悖论其实是无穷级数在作怪, 芝诺的悖论只不过是把1通过二分法分为了无穷份而已, 也就是指级数：
　　无穷级数在数学史上出现的很早. 古希腊时期, 芝诺利用二分法提出著名的芝诺悖论就涉及到公比为1/2的几何级数, 亚里士多德(Aristotle)知道公比小于1(同时要大于0, 那个时代还没有负数的概念)的几何级数可以求出和数, 后来阿基米德(Archimedes)使用几何级数求出了抛物线弓形的面积.
　　到了中世纪，数学家对于涉及到无穷的一些悖论展开了激烈的争论. 1703年由数学家格兰迪(Grandi)发表的格兰迪级数引起了数学家的一番热议, 这个级数即是指：
　　其中丹尼尔&伯努利(Daniel Bernoulli)和欧拉(Euler)也对其进行了研究, 后来经由柯西(Cauchy)引入收敛与发散的定义, 数学家们才知道格兰迪级数是发散的.
　　调和级数也是一个发散的级数, 它的知名度甚至比格兰迪级数还高, 当然它的敛散性也曾是数学家的一个热议话题, 奥雷姆(Nicole Oresme)的对其发散性的极其简洁的证明曾引起数学界的一时轰动, 他是这样证明的：
　　调和级数的通项都是趋于0的, 似乎它应该是一个收敛的级数, 但是它却是一个发散的, 这看上去也像是一个悖论.
　　如果学过数学分析或者高等数学, 应该会对级数的敛散性有所了解, 并对收敛级数的性质有着较好的把握, 所以我们会把重点放在发散的级数上, 对发散级数做一些必要的探讨. 虽然柯西之后的多数数学家遵循柯西摒弃了发散级数, 但是欧拉在不考虑收敛性下通过对级数神乎其技的变形而得到的美妙结论, 以及后来发现的发散级数在渐进估计中的重要应用, 这些都说明了研究发散级数并不是没有意义的. 另外柯西本人也注意到了ln(Γ(x))亦或是ln(m!)的斯特林(Stirling)级数(这个级数是发散的)在计算阶乘时逼近的优点, 但他并没有对这件事作出合理的解释.
　　本文主要介绍发散级数在非柯西意义下的求和问题, 并将会特别考虑格兰迪级数的求和. 通过格兰迪级数在其他意义下的求和, 来说明柯西关于级数收敛的定义并非是绝对的.
　　在行文的最后我们会讨论与格兰迪级数相似但也很具有特殊性的0-1级数. 笔者在此着重于对发散级数求和方式的介绍, 并不刻意追求严谨, 所以对于给出的结论也都是采取“只给不证”的方式，希望这篇关于发散级数的科普文章可以激发读者对发散级数的兴趣, 以及对于发散级数的思考和研究.
　　下面的内容，将以截图形式展现。
　　为了更加深入的比较切萨罗可和与柯西可和的区别, 下一篇会先转入对级数的权的讨论, 但之后还会继续讨论切萨罗和。
　　下次再会。
　　关注微信：DuoDaaMath每天获得更多数学趣文
欢迎举报抄袭、转载、暴力色情及含有欺诈和虚假信息的不良文章。
请先登录再操作
请先登录再操作
微信扫一扫分享至朋友圈
哆嗒数学网站长开辟的专栏。
中国高校校报协会副会长......
北京教育音像报刊总社评论部评论员.....
中国青少年研究中心首席专家
美国独立教育顾问协会认证顾问
中国人民大学政治学教授All material provided on this website, is copyright protected
Custom Search
| For Toddlers...  ~ A gathering place for moms who have young children ages baby-48 months.~ The #1 read post on my blog! ~ Links to our Tot School posts sorted by ages. ~ Montessori inspired activity trays for tots.~ Free themed printables for toddlers {and preschoolers} ~ A 26 unit printable program for toddlers.~ Another A-Z printable program for toddlers {or preschoolers} based around animals.For Preschoolers...     ~ I developed this Bible based printable preschool program for my son and now share it with others!  The base program is free, I offer a , and additional
that go along with the units in the Members Only PowerPoint section. ~ Here you will find
based around fun themes. ~  Here you find free printables for preschoolers that are focused on specific skills, such as ABCs, 123s, Colors, and Shapes.For Kindergartners... ~ This is the extension of
but focuses on skills for the Kindergarten child.  It is a free printable Bible based program for Kindergarten. ~ This is a printable sight word program, focused on pre-primer and primer Dolch Sight Words.  You can download most of the units for free, or in a bundle.  ~ Here you will find free printables for Kindergartners focused around fun themes.  Many of the
are also appropriate for Kindergartners also. ~ A collection of fun and active ideas all centered on learning to read!For Older Children... ~ posts about 3rd grade and beyond {which it out of my early childhood comfort zone} ~ Our collection of Lapbooks we have done, and Lapbooking printables I have created.~ Printables for your calendar with a specific theme.  Target grade 3-6.For Christian Families... ~ Ideas for putting together your own family Bible time at home. Links, resources, and blog posts showing what we did! ~ Free printables to use with your family to reinforce the Bible verses from the AWESOME .Looking for Seasonal or Theme Ideas?~ A collection of blog posts highlighting specific printables and ideas I have shared within a theme. {, , , , , , etc.}~ A collection of crafts and other special activities we have done. ~ I have TONS of theme boards on Pinterest!Looking for Gift Ideas or Learning Toys? ~ A general collection of toys and items we love! ~ Birthday and Christmas gift ideas for toddlers. ~ Birthday and Christmas gift ideas for preschoolers. ~ Birthday and Christmas gift ideas for school age boys.~ Birthday and Christmas gift ideas for moms. ~ A blog where I share links to many of our own personal school toys and tools.... Animal ABC Bundle, RRSP Bundle, You Can Read Bundle, Members Only PowerPoints, and .  A HUGE majority of what I offer is totally free, This post explains the few things I offer for sale! ~ A list of posts I have done about specific areas....the #1 questions I receive!  It is 99.9% on the end of the user and everyone I know has solved their issue with tips in this post! ~ A detailed list of everything we are currently using.    
   Visit the
to learn more what we are all about!What's the difference between the blog and the website?The
is where I update daily, you can
to receive free email updates.  Printable files I offer are hosted here on the website.  Use the navigation bar over on the right to see different categories!Looking for something?  Try searching below with keywords, if this search box doesn't help you, try the search box on the .  It searches the blog specifically!