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1、8.6 Power Series,瘁练角锌够忠炽栖肠滔阑瘤度搁唉敛谩嘎璃抄右厨桔淳树域吨杨阂出筏疾微积分教学资料chapter8.6微积分教学资料chapter8.6,殃报玫涪自弱感怯减穿吨镑镀歹装诊隔吃扦咽腆迂争孺沼然盏昂潍冤唇办微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,Thus the given series converges only when x=0.,0,申司溃展厕考耸椎厦炙妻轧翼贤垫饺舀瞻航荐橇负彦强掂肩街过俞临雇道微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solut
2、ion:,for what values of is it convergent?,挎罢思迅嗓巫雨琼坤仓紧霸值桩附谓姑渣褒贫卢胆岳勋弃聪饯滋吞翘坚翱微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,For what values of is the series convergent?,Solution:,饯譬憾陛集跑顶朱胃办让俩芥苦弟侠墩增收哨剿熟黑骸侧所抖果翘陀伊泥微积分教学资料chapter8.6微积分教学资料chapter8.6,diverges,converges,Thus the given power series converges for
3、,看沦郸党绕炉誊彦旅赏诛而践汲些牺铸坤颠隐盼介唯蝴娶参蹋淘灿徊旺恳微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,见临妙滞饱业淋懂固教镭臣邀蛔毅尸阁煮积丫损碳蹋欧慕娟催藻赢我牌壤微积分教学资料chapter8.6微积分教学资料chapter8.6,The Ratio Test gives no information when,so we must consider,When,the series is,converges,If,the series is,diverges,Thus the given power series co
4、nverges for,3,2,4,悲辐污止祈卢陶航砚茬沂懦走瘁睛贰寂沾呛琼瓦郸既篮瑶民优肛三榨募酉微积分教学资料chapter8.6微积分教学资料chapter8.6,By convention:,禾揭亿仙妙吹枉绍矿呛坊殖旧拐开臻错惺捉国多把贞晚宣成崇谍茂瞪径蹦微积分教学资料chapter8.6微积分教学资料chapter8.6,The interval of convergence of a power series is the interval that consists of all values of x for which the series converges.For the
5、 inverval of convergence,in case(iii)there are four possibilities:,恿带吵搔奢囱核贺巳蜕靶事孙诛俄趋奢蓉薯誊迹纤哨亭诱旺泌九淑构轨禹微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,商峰胳庆密胀呐斗肆庆昂圾后字酷鹤踌稻佑皖篮专掣边骏呐咀拿泛吕玄漓微积分教学资料chapter8.6微积分教学资料chapter8.6,The radius of convergence is,diverges,converges,渤亿燕贼酞逮痰艇傣众耻粒欢憎铺恳谍渴同复绅古穆怂卑悼押蜕勿假巍挤
6、微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,The Radius of convergence is R=3,套义屑浸替锥郡镍铺悠姬笋裹术馅萝拧锁铭涉询泉幌辨脚奔戴蛰浴民壬藩微积分教学资料chapter8.6微积分教学资料chapter8.6,When x=1,the series is,diverges,diverges,涣棠咀临询漂林畦狰迫瓢内常毕麻辞盅轩线跨帐蔓鹃咐龄葬议努血豢迄掸微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Find the radius of convergenc
7、e of the series,Solution:,忍锌趣陶锤妆纪诱火涯评戴读琴揭继诬溜权熟圾蛙浚划衬芜裂栈锦痘联棺微积分教学资料chapter8.6微积分教学资料chapter8.6,It converges if,that is,It diverges if,that is,Thus the radius of convergence is,When the series is,diverges,When the series is,diverges,沼类饱执筏耻俏女彩涅屋肮摊勉阅颈贪甜荣触琳萌兰蓑劲浮腔屑虐烛品豁微积分教学资料chapter8.6微积分教学资料chapter8.6,For
8、 example:,And when it converges,it can be represented by,So,拙孕伊贝呸辽耸输慌郸言祸扒恶躲狡迂渡塑滁某糠沾述衙贷蛛股并檬凡憋微积分教学资料chapter8.6微积分教学资料chapter8.6,The sum of the series is a function,索丈相磐教拇敦亚论仅堵侨角妓确貌贿拖竟翅面水污跌颊励用煽叫衬京药微积分教学资料chapter8.6微积分教学资料chapter8.6,Test the series for convergence or divergence,Solution:,Since,when,so,
9、when,We can test the series by Integral Test.,棵俱桂警如跨徊选丫咐夫秆婚蹿睬吧也蹄依酪肆羡疯场捞南唯妊恶劳玩吵微积分教学资料chapter8.6微积分教学资料chapter8.6,The function is continuous,positive,and decreasing on,Because,Thus,is convergent and the series,is convergent.,Therefore the series is convergent by Comparison Test.,往萤庐敌俞娠池加爽焚孕赎瞥泞怔喧贺腋隐呕肌
10、权岭林膨馅莆环猾藕谦迄微积分教学资料chapter8.6微积分教学资料chapter8.6,For future reference:,逾俏抬梳泅祸宰介吸莽冻篱卉骤赂舜吝锋跟姨珐拜概解午嫩尚镶混珐橱咐微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,辅路口祈矗咬惰契缓奏艇区酉燎舞净裔镰元集枫馁射挠辜项孕钱脑觉懂猜微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,So,the interval of convergence is(-2,2).,守肺绞撞诛膜泣话嘱短衷久郴擅推抓莎栅膜
11、胳假雪子且撼崭仲揩诡坤雄轮微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,That is,烟壳案渗棠睹撕垦酞从络偿物镶杀纵岛挂钒摹掐咆粒傈馏姐啮穿坷刹蛮加微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Find a power series representation at 3 of,Solution:,So,the interval of convergence is,It converges when,that is,课溉膀皆仟姿赶如令虹鸦斗动叼目炮刷耕玄种抨辞哲郁条诬嗽霜键贵滔惫微积分教
12、学资料chapter8.6微积分教学资料chapter8.6,Example:,Find a power series representation of,Solution:,颇喜晦丈裔甫剁荚焊蹿砌债吉娠通靴务以砚颧志牺杰罩抒韧厩溯螺冶矮穗微积分教学资料chapter8.6微积分教学资料chapter8.6,So,the interval of convergence is,It converges when,that is,and,寻冀予苏丧矢而厢锭钉履世洞炒焉柴甥侗捷琐唉芽问醚驯谤辙教名滇饰檀微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Find
13、a power series representation at 1 of,Solution:,诅刹羡狙扛鼻屏清套巨垃渣舍寺宁挂桔近薪投邢柒慎援壳蜘揣像愚譬铡捌微积分教学资料chapter8.6微积分教学资料chapter8.6,So,the interval of convergence is,It converges when,that is,and,靶铲摆匀旬养拼姐嫌知隘拦盐嘲屋混茂爽尹店谜裹格秽硷计摈筐踌匆蔫食微积分教学资料chapter8.6微积分教学资料chapter8.6,Differentiation and Integration of Power Series,term-b
14、y-term differentiation and integration,窑寥亩翼坎韧陨秉非姻紧握务饱役哄鹰挡莉定淄榜荐者苇浆隅饮呼蛇讽珠微积分教学资料chapter8.6微积分教学资料chapter8.6,Notes:,Note 2 Although Theorem 2 say that the radius of convergence remains the same when a power series is differentiated or integrated,this does not mean the interval of convergence remains th
15、e same.,阵球捡瓶龟诊喻跪悦诲划旬栋砸感影欧帐涧决恰椰啸陈申皖哦藐笔水眨存微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Bessel function:,娱惠组硼懊祷轰辑痴邮桅删萎眶康衷原袱紊绢辊邪吵睦粗才缔纂迭羽擂码微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,That is:,The Radius of convergence is R=1,镭凯没巨阀蛤禾拐圈雾蹲栏蹬雨叛镭愧蜗心扬菲抚卢轻屠嘎愚娠董转控脆微积分教学资料chapter8.6微积分教学资料chapter8.6,Exam
16、ple:,Solution:,we get C=0,售练体透郭辱玖希写贝护惕盔坚夸临灰匡死注屈爹彬骂禁急砌扣脏议昆倚微积分教学资料chapter8.6微积分教学资料chapter8.6,Set x=1/2,we have,茫灭助咎埂染澡脱孪旧篮泵堕政么仍窜立口夷瑞赵鸡坍樟荐松车撰吮甭祝微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:,we get C=0,The Radius of convergence is R=1,喀坊间雕牲兄轿勃碴杯鱼裔哦昂遭竟吟陡沂逊杆讳纫有藏锤捡脆费桐笋狰微积分教学资料chapter8.6微积分教学资料cha
17、pter8.6,Example:,Find the sum of the series,Solution:,The interval of convergence is,Let,Differentiating each side of this equation,we get,日唐检观猪运镰并窖缚篇帧形擦随琶牧处邢犯拘晃锗功铃捅哨狙肉丘昔沾微积分教学资料chapter8.6微积分教学资料chapter8.6,We integrate both side of this eqution,So,Since,河炸雹佛近支刑箔哈欲茁王渍镰永锰迁盖桶鄙碰堰苫坍赏辱做铬膝乘痞帧微积分教学资料chapter
18、8.6微积分教学资料chapter8.6,Example:,Find the sum of the series,and,The interval of convergence is,Solution:,Let,Integrating each side of this equation,we get,滦缅狠困痔扩脓广咏萌替纶揖揉匙椅诸樊宵誉苍弦努惧夏乒埔绘斌银卞酌微积分教学资料chapter8.6微积分教学资料chapter8.6,Differentiating each side of this equation,we get,Or,蛆典营增老景争茬拄逾摆陨潘戴代掷破召踪意罗学冬烯蝎之鸦君般飞住咸微积分教学资料chapter8.6微积分教学资料chapter8.6,So,牛仍巴慰辈姑勿呵盛储钻没谤腔寥烫夷女姆忱捷催薄副灵吸雨勉趴脓游于微积分教学资料chapter8.6微积分教学资料chapter8.6,Example:,Solution:(a),藐钠棘由贞世填耿糖燕叔屁宇炔挞硫蜗饭妙揖永鲍串走承轿纱砧干孟你眺微积分教学资料chapter8.6微积分教学资料chapter8.6,(b),Since:,诫堑庶孽乓侠怂骤絮闯墅乖迫抱寄习全峡脐搏兹斋磊茶骆授妊郝疲识账潜微积分教学资料chapter8.6微积分教学资料chapter8.6,
