本科毕业论文-沃利斯公式的证明及其应用.doc

盐 城 师 范 学 院毕业论文 沃利斯公式的证明及其应用学生姓名 学 院 数学科学学院 专 业 数学与应用数学 班 级 10（2）班 学 号 10211255 指导教师 韩 诚 2014年 5 月 25 日沃利斯公式的证明及其应用摘 要Wallis公式在求Euler-Poisson积分和推导Stirling公式的过程中扮演了很重要的角色近几年来，国内很多数学分析的教材都引入Wallis公式，但教材中关于其应用的论述很少本文针对Wallis公式的证明并将Wallis公式进行两个简单推广，从数列极限计算、积分计算以及级数收敛性判断几个方面探讨Wallis公式的应用，为微积分教学提供有意义的素材和思路【关键词】Wallis公式；极限；积分Proof and Its Applications of Wallis FormulaAbstractThe formula of Wallis plays an important role in the process to obtain the Euler- Poisson integral and the derivation of Stirling formula. In recent years, many domestic analysis mathematics textbooks into Wallis formula, but little about the applications of the teaching material. This paper proves that the little Wallis formula and the Wallis formula is two simple promotion, as well as the series convergence judgment application aspects of Wallis formula from the sequence limit calculation, integral calculation, to provide significant material and ideas for the teaching of calculus.Key words Wallis formula, limit, integral目 录引 言11 沃利斯公式的证明及推广11.1沃利斯公式的新证明11.1.1有限次代数方程根与系数的关系类比到无限次方程11.1.2应用含参量积分证明沃利斯公式31.2沃利斯公式的推广41.2.1含参数的沃利斯公式41.2.2含沃利斯公式的不等式52 沃利斯公式的应用62.1 沃利斯公式在极限计算中的应用62.2 沃利斯公式在积分计算中的应用92.3 沃利斯公式在级数收敛判别中的应用113 总 结13参考文献14盐城师范学院毕业论文引 言近几年来，国内很多数学分析教材都引入Wallis公式，关于其证明方法有很多种，一般都是利用积分证明的，本文将借助类比思维，分别利用根与系数关系的思维方法和含参量定积分来证明Wallis公式此外，教材中关于其应用论述的很少，这是为什么呢？因为很多可以应用Wallis公式的“高地”被斯特林公式占领了但本文搜集到一些不能应用斯特林公式却可以能应用Wallis公式的例子且Wallis公式在推导斯特林公式中扮演很重要的角色，从加深理解Wallis公式的角度探求其一些简单推广以及其在极限计算、积分计算和级数收敛判别方面的应用1 沃利斯公式的证明及推广1.1沃利斯公式的新证明沃利斯公式指的是 . 经过开平方后，则Wallis公式可以写为 . 现引入这样的数学记号：，则Wallis公式又可以写成. （1-1）1.1.1有限次代数方程根与系数的关系类比到无限次方程类比的思维是人们把个别问题解决后所得到的经验用来解决其他近似问题的一种类似联想的思维的方法，类比这个重要的数学思想方法，曾被波利亚称为科学发现的“伟大引路人”，被17世纪德国著名天文学家和数学家开普勒视为“知道大自然一切秘密”的“导师”.在这我们也将采用类比思维.对于有限次代数方程， 假如有个不同的根，那么左边的多项式就可以表示为线性因子乘积，即 比较这个恒等式两边的同次幂的系数，就可以得到根和系数的关系.特别是偶数次方程有个不相同的根，则有，我们比较二次项系数有. 根据幂级数展开式，在，则. 利用无穷多项方程. （1-2）由于方程（1-2）的根为：，则即. （1-3）因为绝对收敛，所以这无穷乘积是绝对收敛的. 在（1-3）中令，得,沃利斯公式（1-1）得证.1.1.2应用含参量积分证明沃利斯公式引理1 设，则有 . 定理1 设，证明.证明 令，根据引理1得 .由于，因此当时，即.当时，.则 另一方面，由定积分的保不等式性质知，当时，有,从而得到，从上式可得到.在上式中，令，则.由于，因此根据迫敛性可知，因而.Wallis公式（1-1）得证. 1.2沃利斯公式的推广1.2.1含参数的沃利斯公式对任意非负实数和正整数，则有 （1-4）其中. 证明 由分部积分法知，当时，则有 .因此有.于是 ，从而，即，令，利用夹逼定理并整理得到（1-4）式.注1 令，可以得到著名的Wallis公式1.2.2含沃利斯公式的不等式关于Wallis公式的研究一直以来都受数学家的关注，1956年给出了如下含Wallis公式形式的不等式本文将含Wallis公式不等式推广为 当时，有下列式子成立 , （1-5） 或 . （1-6）证明 如果，式（1-5）显然成立.如果，用数学归纳法证明，式（1-5）左边当时，显然成立.假设对式子（1-5）的左边对于正整数n成立，则下面证明对于同样成立，由归纳假设，只要证明,即证明 ,亦即 . （1-7）根据伯努利不等式 .令，则.所以式（1-7）成立.因此，对任意正整数，式子（1-5）的左边成立.下面证明式（1-5）的右边成立.当时，要证明(1-5)的右边成立，只要证明,即可，化简可知这个不等式成立的充要条件为，又由于时，有.因此，此时式（1-5）的右边成立 .假设式（1-5）的右边对于正整数，下面证明同样成立，只要证明,而此不等式成立的充要条件为,即 （1-8）但是，由Newton二项式公式，式子（1-8）的右边不小于下面的式子：.所以式子（1-8）成立.因此，对任意正整数,式子（1-5）的右边成立.2 沃利斯公式的应用2.1 沃利斯公式在极限计算中的应用由于沃利斯公式和极限有关，所以在计算一些极限的问题可以通过沃利斯公式会很容易出来.例1 求极限.解 利用沃利斯公式（1.3），可得 .例2 设（），试证，.解 由于， 因此，是递增数列.根据沃利斯公式，则， .得证.例3 求极限.解 由沃利斯（Wallis）公式的推广（1-4），则有 .令则 .例4 求极限.解 因为 ，.因此 （2-1）由于当时，级数在处收敛（本文下面给予证明），又由于函数项级数M-检验法知，级数(1)在上一致收敛.在（2-1）中，令，有， （2-2）对（2-2）所在区间取积分，并且由逐项积分公式，则有，又由沃利斯公式可知，于是 即.2.2 沃利斯公式在积分计算中的应用对于一些不易用积分法求出原函数的积分，但是利用沃利斯公式却可能很容易解决这些问题.例5 求积分.解 假设，由,可知 ,注意，前者仅对正确，而后者对任一都对，由此可得 ， .取积分.但用替换可得.又,即,所以.平方得.由沃利斯公式得.可知，当时,即.因此.例6 求的值.解 .例7 求积分的值.解 令,则.原式 . 2.3 沃利斯公式在级数收敛判别中的应用对于一些级数收敛性的判别问题，文献10指出若利用沃利斯公式可能会起到事半功倍的效果.例8 判别正项级数，（）的敛散性.证 由于通项含有双阶乘的运算，原则上想到运用比式判别法，但是由于，因此比式判别法失效.若运用拉贝判别法，由于 ，所以当时，即时收敛，当时则发散，但当时拉贝判别法则无法进行判别.但如果利用沃利斯公式，不仅对于和时的情况可以判别，而且对时的情况也能判别.比如：由沃利斯公式得 .则正项级数与正项级数的敛散性相同.由上分析可得正项级数当时收敛，当时发散 .例9 二项式级数 当，时条件收敛 .证 令，表示二项式级数在时的通项，则 ，故此二项式级数是一交错级数，且 ，由于，则必存在两个正整数和，使 ，再结合沃利斯公式的推论中式子（1-5）可得.即，由判别法可知级数收敛.又沃利斯公式推广中公式(1-6)得 .再由调和级数发散可知级数发散.所以当，时二项式级数条件收敛.3 总 结本文针对沃利斯公式的应用进行研究，给出了沃利斯公式在求某些极限计算、积分计算、级数收敛的简便之处.并且将沃利斯公式进行简单的推广，在证明某些级数收敛性问题时，运用达朗贝尔法与拉贝判别法时会失效，但运用沃利斯公式会很简单有效的解决这类问题，此外我们知道在有关二项式级数在收敛区间端点的收敛性是一个较为困难的问题，有的教材对此置之不理，有的则要借助于几何级数来解决，本文利用对沃利斯公式的推广能有效的解决一些此类型的问题.当然还有更多问题值得我们探讨，例如对含参数的沃利斯公式的更多应用以及含沃利斯公式的双边不等式的推广可以给出更为精确的结果，以及沃利斯公式在二项式的上下界的研究等，这些问题将另文研究. 参考文献1 华东师范大学数学系数学分析上册（第四版）北京：高等教育出版社，20012 屈芝莲.Wallis公式新证明.科学技术与工程,2011,1:549-550.3 Mikhail Kovalyov. Elementary Combinatorial-Probabilistic Proof of the Wallis Formulas. Journal of Mathematics and Statistics,2009,5:408-409.4 李建军.一种含参数的Wallis公式与Stirling公式.数学理论与应用,2008,3:52-53.5 赵德钧.关于含Wallis公式的双边不等式.数学的实践与认识，2004,34(7):167-168.6 D.k Kazarinoff. On Wallis formula. Edinburgh Math Notes,1956,40:19-21.7 张文亮.一个不等式的探讨.2004,3:19-21.8 Jeffrey H. Wallis formula for .Methods of Mathematical Physics,1988,3:468-467.9 王振芳,陈慧琴.沃利斯(Wallis)公式及其应用.山西大同大学学报,2011,10:5-6.10 张国铭.Wallis公式的几个应用.高等数学研究,2008,9:37-40.请您删除一下内容，O(_)O谢谢！Many people have the same mixed feelings when planning a trip during Golden Week. With heaps of time, the seven-day Chinese请您删除一下内容，O(_)O谢谢！National Day holiday could be the best occasion to enjoy a destination. However, it can also be the easiest way to ruin how you feel about a place and you may become more fatigued after the holiday, due to battling the large crowds. During peak season, a dream about a place can turn to nightmare without careful planning, especially if you travel with children and older people. As most Chinese people will take the holiday to visit domestic tourist destinations, crowds and busy traffic are inevitable at most places. Also to be expected are increasing transport and accommodation prices, with the possibility that there will be no rooms available. It is also common that youllwait in the line for one hour to get a ticket, and another two hours at the site, to only see a tiny bit of the place due to the crowds. Last year, 428 million tourists traveled in China over the week-long holiday in October. Traveling during this period is a matter that needs thorough preparation. If you are short on time to plan the upcoming Golden Week it may not be a bad idea to avoid some of the most crowded places for now. There is always a place so fascinating that everyone yearns for. Arxan is a place like this. The beauty of Arxan is everlasting regardless of the changing of four seasons. Bestowed by nature, its spectacular seasonal landscape and mountains are just beyond word. Arxan is a crucial destination for the recommended travelling route, China Inner Mongolia Arxan Hailar Manzhouli. It is also the joint of the four prairies across the Sino-Mongolian border, where people gravitate towards the exotic atmosphere mixed with Chinese, Russian, and Mongolia elements. As a historic site for the Yitian Battle, Arxan still embodies the spirit of Genghis Khan. Walking into Arxan, you will be amazed by a kaleidoscope of gorgeous colors all the year round - the Spring azaleas blooming red in the snow, the Summer sea wavering blue in the breeze, the Autumn leaves painted in yellow covering volcanic traces, and the Winter woods shining white on the vast alpine snowscape. Hinggan League Arxan city is situated in the far eastern area of Inner Mongolia Autonomous Region. Its full name Haren Arxan means hot holy water in the Mongolian language. Arxan is a tourism city in the northern frontier with a blend of large forest, grand prairies, vast snowfield, heaven lake cluster, thermium, as well as volcanic cluster. It is a rare and unique ecotourism base filled with healthy sunshine, clean air and unspoiled green. Nestled close to the countrys largest virgin forest, and known for its spring and ecological environment, Arxan is marveled at by many tourists as the purest land on earth. You cannot miss out the Autumn of Arxan. It is definitely the best with brightly-colored scenery full of emotions. Autumn in the northern part of the country comes earlier than the South. A September rain followed by the footprints of Autumn brings more colors to the once emerald green mountain and blooming grassland. Shutterbugs flock to see for themselves the marvel of splendid colors around the mountains and waters, many of whom have travel a long distance and even camp here only to capture a moment of the nature wonder. The silver birch turns golden, while the larch is still proudly green. You will find yourself drowned in the intoxicating red of the wild fruits as well as the glamour of flowers in full blown. And your heart will be lingering on the woods as its time for the wild fruits to ripe. The picturesque Arxan in Autumn is indeed a fairyland only exists in a dream that satisfies all your fantasies. If itrains heavily on Saturday night, some elderly Chinese will say it is because Zhinu, or the Weaving Maid, is crying on the day she met her husband Niulang, or the Cowherd, on the Milky Way. Most Chinese remember being told this romantic tragedy when they were children on Qixi, or the Seventh Night Festival, which falls on the seventh day of the seventh lunar month, which is usually in early August. This year it falls on Saturday, August 2. Folklore Story As the story goes, once there was a cowherd, Niulang, who lived with his elder brother and sister-in-law. But she disliked and abused him, and the boy was forced to leave home with only an old cow for company. The cow, however, was a former god who had violated imperial rules and was sent to earth in bovine form. One day the cow led Niulang to a lake where fairies took a bath on earth. Among them was Zhinu, the most beautiful fairy and a skilled seamstress. The two fell in love at first sight and were soon married. They had a son and daughter and their happy life was held up as an example for hundreds of years in China. Yet in the eyes of the Jade Emperor, the Supreme Deity in Taoism, marriage between a mortal and fairy was strictly forbidden. He ordered the heaven troop to catch Zhinu back. Niulang grew desperate when he discovered Zhinu had been taken back to heaven. Driven by Niulangs misery, the cow told him to turn its hide into a pair of shoes after it died. The magic shoes whisked Niulang, who carried his two children in baskets strung from a shoulder pole, off on a chase after the empress. The pursuit enraged the empress, who took her hairpin and slashed it across the sky creating the Milky Way which separated husband from wife. But all was not lost as magpies, moved by their love and devotion, formed a bridge across the Milky Way to reunite the family. Even the Jade Emperor was touched, and allowed Niulang and Zhinu to meet once a year on the seventh night of the seventh month. This is how Qixi came to be. The festival can be traced back to the Han Dynasty (206 BC-AD 220). Traditionally, people would look up at the sky and find a bright star in the constellation Aquila as well as the star Vega, which are identified as Niulang and Zhinu. The two stars shine on opposite sides of the Milky Way. Customs In bygone days, Qixi was not only a special day for lovers, but also for girls. It is also known as the Begging for Skills Festival or Daughters Festival. In this day, girls will throw a sewing needle into a bowl full of water on the night of Qixi as a test of embroidery skills. If the needle floats on top of the water instead of sinking, it proves the girl is a skilled embroiders. Single women also pray for finding a good husband in the future. And the newly married women pray to become pregnant quickly. Tradition transformed The love story of Niulang and Zhinu, and the Qixi Festival have been handed down for generations. Yet these ancient traditions and customs are slowly dying out. Many modern Chinese, particularly youngsters, seem to know more about St Valentines Day on February 14, characterized by bouquets of roses, chocolates and romantic candlelight dinners, than they do about their home-grown day for lovers. Even Qixi is nowadays referred to as the Chinese Valentines Day. More and more young Chinese people begin to celebrate this day in a very similar way as that in western countries. Fewer people than ever will gaze at the heavens on Saturday to pick out the two stars shining bright on either side of the Milky Way, that is, if people even know on which day Qixi falls. There are ready reminders dotted about, in the form of big ads saying Sales on Chinese Valentines Day! in shops, hotels and restaurants. But few young women will mark the festival with their boyfriends, or take part in traditional activities to pray for cleverness. A large-scale of exhibition on the Silk Road opened to the public in the National Museum on Thursday. The exhibition displays more than 400 treasures from Shaanxi province, Gansu province and 14 other regions. It is largest scale exhibition that shows the history of the Silk Road in recent years. AsAPECis going on in Beijing, the exhibit aims to show the world this brilliant section of history. The Silk Road is a series of trade and cultural transmission routes that were central to cultural interaction through regions of the Asian continent. It connected the West and East by linking traders, merchants, pilgrims, monks, soldiers, nomads, and urban dwellers from China and India to the Mediterranean Sea during various periods of time. I met my dear hubby in 2009 the second year I was in my school and it was the first year when he came here. We became colleagues. And there was nothing special happening during that year. He was just a kind and humble brother graduated from the same college as me. But we did not know each other till he became my colleague and his office desk was just in front of me. There was only a glass between us. So actually we could see each other when we both look up. But it is not the main reason that really draws us close. How we become close to each other was something seems quite natural. We went out to play badminton with other colleagues and we sometimes chatted for a while. He is a man of few words but somehow sometimes will say something really humorous. Then one day, to my great surprise, I found a pot of beautiful flower on my desk. It was just the flower I had talked about with my colleague the other day, and I thought it was her who put the flower there. But the truth is not. It was the humble guy sitting in front of me. He just simply said ,”well, I caught sight of this flower and thought that its beautiful and you may like it”. I wonder why I was moved because of these simple words and present. Maybe it is the sincerity and the true self that impress me. I thanked him and some colleagues walked past my desk would appreciate the beautiful flower and they noticed that something romance happened there. Then later on, there were some little surprises once in a while. Once, there was something wrong with my drawer and I had complained about it unconsciously. And he fixed it for me. Then the other day, he placed several pink diamond roses beside the flower with a piece of paper saying” I love you more than I can say! Diamond stands for marriage, rose stand for love. I hope you are mine, become my wife. You are the only girl who I really want to spend the rest of my life to stand by.” Although there were some grammar errors but I got what he meant. He meant so clear and so direct. And actually, I accepted him some days earlier. I was just waiting for him to break the ice. So finally, the common interests and the common understanding brought us together. Some people say the relationship and the feeling will change after getting hitched. You will find that love is no longer strong and there is always lack of passion and whats worse you may find out many bad habits of your partner, which never appear when you are first in love before the marriage. It seems that many faults turn up after you truly living together and see each others face the moment you open your eyes in the early morning. But after almost 4 years from now, I do not feel any negative change in our re