2020年高考中的一个不等式的探讨

数学星空

设\(n \in N,\theta\in R\), 求证\(\sin(\theta)^2\sin(2\theta)^2\cdots\sin(2^n\theta)^2\leq (\frac{3}{4})^n\)

有网友给出了很简洁的解答：

设\(f(x)=\sin(x)^2\sin(2x)\),上面不等式左边记为\(m\),则有

\(f(x)=\sqrt{4\sin(x)^6\cos(x)^2}=\sqrt{\frac{4}{3}\sin(x)^2\sin(x)^2\sin(x)^2 (3\cos(x)^2)} \leq \sqrt{\frac{4}{3}(\frac{3\sin(x)^2+3\cos(x)^2}{4})^4}=(\frac{\sqrt{3}}{2})^3\)

\(m^{\frac{3}{2}}\leq \sin(\theta)^2(\sin(2\theta)\sin(4\theta)\cdots\sin(2^{n-1}\theta))^3\sin(2^n\theta)=\prod_{k=0}^{n-1}f(2^k\theta)\leq (\frac{\sqrt{3}}{2})^{3n}\)

现在我们的问题的是：如何更精确的给出左边的估计，我们可以记为

\(S(n,\theta)=\sin(\theta)^2\sin(2\theta)^2\cdots\sin(2^n\theta)^2\)

可以利用数值计算或者渐近分析或者其它的计算方案

mathe

取极值条件为$\sum_{k=0}^n \frac{2^k}{tan(2^k\theta)}=0$,可以利用倍角公式转化为$tan(\theta)$的$2^n$方程

mathe

比如n=2,我们有
? g(x)=2*x/(1-x^2)
%1 = (x)->2*x/(1-x^2)
? 1/x+2/g(x)+4/(g(g(x)))
%2 = (2*x^4 - 9*x^2 + 3)/(-x^3 + x)
? polroots(2*x^4 - 9*x^2 + 3)
%3 = [-2.0340743862547621747212413489552276777 + 0.E-38*I, -0.60211410146442557444368645010866685442 + 0.E-38*I, 0.60211410146442557444368645010866685442 + 0.E-38*I, 2.0340743862547621747212413489552276777 + 0.E-38*I]~
所以在$\tan(\theta)=2.0340743862547621747212413489552276777$或$\tan(\theta)=0.60211410146442557444368645010866685442$时取极值
对应$\sin(\theta)\sin(2\theta)\sin(4\theta)$为-0.68730469339345227373910004195465128424或0.30604149364155013456189465119387773308

mathe

取极值时$tan(\theta)$满足的方程

n	方程	极值点$tan(\theta)$	$\frac{2^n(\theta -\frac{\pi}3 )}{\pi}$	极值	极值的代数表达(方程的根)
2	2*x^4 - 9*x^2 + 3	2.0340743862547621747212413489552276777	0.084892363556764800725766656117986720779	0.47238774156066743757961292457432515419	55296 - 506088 x + 823543 x^2
3	3*x^8 - 49*x^6 + 129*x^4 - 55*x^2 + 4	1.6282690413480532424830358018056910865	-0.069161306041355184275941745556086179678	0.34171462946895103514328399547996040900	29686813949952 - 3205444645421056 x + 50799644415568256 x^2 - 271588578052757152 x^3 + 437893890380859375 x^4
4	4*x^16 - 253*x^14 + 3531*x^12 - 15189*x^10 + 24819*x^8 - 15631*x^6 + 3769*x^4 - 287*x^2 + 5	1.7934808484460177471664012026666617814	0.076182860324977237909034452102919069995	0.26047330635733330291786453109181926013	5640364303994013877509134745600000-23651613724692446931595236613359140864 x+5436219938052688645553470979951204237312 x^2 -445584843177218246190539235836898226208768 x^3+15687831085121786623025290178027600406085632 x^4 -273115779541031417749846624226250522827341824 x^5+2447168371531086836758953887791990904097289728 x^6 -10611014779724847903754555725560151744652894848 x^7+17069174130723235958610643029059314756044734431 x^8
5	5*x^32 - 1229*x^30 + 77131*x^28 - 1837593*x^26 + 20869723*x^24 - 126714521*x^22 + 441953695*x^20 - 923306757*x^18 + 1180658289*x^16 - 930130167*x^14 + 448777105*x^12 - 129870011*x^10 + 21641065*x^8 - 1936483*x^6 + 83269*x^4 - 1383*x^2 + 6	1.7039284017193390606838927894626908967	-0.072494703301028546917166261219525917698	0.19370182814862643535859755476406248091	32640111628754440955977358053698508963062193439825910040528449820052108083200000 -20828822948788775921283939944889544832523688890607300528685739839480549531431829766144 x +187256198708888953952606681309328637759820298851936057004604287435665023835584190880940032 x^2 -511933401845263682994265188311325745337492624376769248951309802415732947206941391847616937984 x^3 +415258061621539394571738071120228639358068399095777423540339752599389683857857843714617864355840 x^4 -164343131795738999950016863862456113624473729392588612098833696739031945824751067403382867695239168 x^5 +37479746741245383868606479120801249804431704579915396055454424041019207094478979949490006748106326016 x^6 -5291769700752004927336443096166925193988296467522768026839270643971849120479683008095924423245630013440 x^7 +478353811564849606124421678724069128636777846174491661641645411417545385758293356638988581140341358329856 x^8 -28169571482269324582723204497606169027596241138600394623910019379288766825350836909108597113281460744224768 x^9 +1093123358693155739858574078053110099877755848386292479558345447455220599027772260317197439130028817160077312 x^10 -28172749993529321426207254660846466088983763685471619519122276651966259465217144050440947505372080824614125568 x^11 +481456541667376573129303672738104528589858343410452511898234073652671744607277457170050933166281687008927940608 x^12 -5354777103530668470542612080988551058319424178930310930804845687430512564162311660822720530793287534175616434176 x^13 +36945581140541149037312111421751689277801956279361791811999298234576010673997450765738009148050689011559712634880 x^14 -141987568474750480629969970869394368970906647490255251334477948466069512339098569254310444434387877400081121987072 x^15 +228273036346967044979900512337165522400819024722490933829954793073267717315004135590642802687246850771579138342847 x^16
6	6*x^64 - 5725*x^62 + 1501891*x^60 - 163794737*x^58 + 9298413447*x^56 - 311176203277*x^54 + 6654895878563*x^52 - 96128567111969*x^50 + 975765693341183*x^48 - 7167164993039601*x^46 + 38948059624730191*x^44 - 159265205234957589*x^42 + 496447188773327555*x^40 - 1191172519726329681*x^38 + 2215696804481587135*x^36 - 3210557673455433333*x^34 + 3634194881023859879*x^32 - 3216624420038640279*x^30 + 2224168387548227465*x^28 - 1198134994878264851*x^26 + 500420894103944213*x^24 - 160915415089864839*x^22 + 39454123980235081*x^20 - 7281860429763651*x^18 + 994825867957133*x^16 - 98415788065883*x^14 + 6848540613413*x^12 - 322364565735*x^10 + 9719028577*x^8 - 173382107*x^6 + 1620373*x^4 - 6375*x^2 + 7	1.7467305657729733109614777242073170109	0.074290831900898337020180493526800342321	0.14587945612387438221188215375808117720	11100735362250400146943985471771535032418248194816906913377233113675861690069003299338162630976644091339090599537000364744055662996724857828212625399758627264052280835440640000000000 -4164230797364082661054472555101083286296404702318440000746111799191099692601150453982123412980475339968335819175159914228122429617233547682108857766018466556670315360023943856965324208340992 x +5715194476004878093932354782853967607961337737869812308948472118874519254765550087679490599296750857596254487507254540146601922941962852799644460307427440134640161384116025585813911345416107982848 x^2 -2083673880677710495821103507444663905839744556453055315117432900019162061076669208714672659685866087500522134991776019402071546379887511018100845173873195984778516750146752638445855961145496393928933376 x^3 +86286813363600434357057716968533805467767952377983924896170715716983988260676333824087806549363751850865723606744077335565398750684715486658779103894722981472203889219260317602853680827183967149377064009728 x^4 -1563574179661471347106463769892236486400161532945511767448828763594360983275247995402863599882684583489474334682881067069040068115981483474655137475784985384708767580087160987797763605740348182095312780116623360 x^5+15610101946715616566563071137598291770699624112619371826559837609780059780324891230592566144135872892354533814779020657066670422490169012540504305259900006001072299819282700086347650053616722813067158758782916362240 x^6-93583923745902635296677353090843872640194085261133377444749631905476767200470030000716299973053946363289324454801202566419607187547895720188200284225058154086559425603515397028336391370363308943557529896706251847368704 x^7+350380505116670938070754277457986482104802773128618303266806991578005112642880238075197822086518801878963696507529062340848670851868695497394062412984017530384227429248055571105520273565685084008761945914189805873585455104 x^8-840683529919252278151483756943403992470163866760942571055729559540744244299322164741040520755303056674858232762321629638728813924707714060682226169203319902155869420941649483288657870135981283048272505485436892063354658488320 x^9+1330321877793867282826433936956654654999738235012384733076197372137183756690884031582674668032481461651643976489667044529525353854982520074030115495630577101985846723588615624042675330278661453360075101569299320238747545174540288 x^10-1432989529681392148755555316451864609304941023914339734782175230998976275780555364632327522737480556971355718043839007985412917072831634696455023288356761863438428347166745717514374148687373978136751034268929229414430366909404807168 x^11+1085183974285515304339594331954947956542789139831220315540804824593898653113873468303864242053375653978815407223882894210649461186843838172202747700567431369784366489813649162396753670122171255743674465657335070746143669049960270135296 x^12-595695215494132422519989827029637379740123674776632023802162227923440589753420484981157460397660135543321361911973380996002707828246011828899120079185101773491438917195462688262135669426440911755986939548722522445529382727137863407763456 x^13+243285099738876088642128388263067744696338153129470513879501433968367331037461910908524680326985097793366294858227249190367619555925575434396509334028969687319601780031885496812906031594798940584803247325811035219292449872466668295022444544 x^14-75462446625738664013312840632345370159757494586827196844850338805051875434619190864568187088445302024158411438846167623682055417829492151878802823187621141926125172950720479284233098708859420511356314082653163549789009245247759756695758176256 x^15+18056741680315100437389416225843454699105495635985013827982639242964513442530572193773189833180840401967414366884228960921069806810756954919354662787356421553126238652090359448159804298310813263817595198222772346185372026977981463210750558339072 x^16-3371168135871938753079147968272414682574868218805471486622858187592591850421073608566075692094695482252582526405997281258673466289037540633036053670342799500807190950867763799204536893697236454713111187359728998120888952129774517500140441783238656 x^17+495022165952031784382767815684522546420845668752523741085295658309942700679499390199837463137351066117408567073635680787164520207173528923400929576617378498573201980486140531910032678322824958481052018506552580830571109093811411961701281794686451712 x^18-57475082613887396410244006232175294971554709171559232251129237164558048714296337074271735089440940550681237340657176381544688413807321297481630517236527538094179540313609848838938460476160674760410513775769647790559979919347893306517662956136651816960 x^19+5293197866489631309204501698867226926203179689013815217714520232243157160991114363200875838900451257177526952935995003703912657578545705086599026372162345843723173932693022552040647061710268254133932667822210882136171119039289533627798331266172188622848 x^20-387185225460803375185182203857532023671324378524956513713777249674054850969836247133443551122206750168096785819853165600956053299935286510632056657450019394571768180095641523168244645873880912969708304745969326295989747374734178508351392001467054083276800 x^21+22486006107511845825885748752579360695062752739084887699914694250917724844877742422501462031351535424695575874245463745204821262942040922188939563102713079904371522191083725834199711957063195208190634509940538667833669073892259605377192515153684676544036864 x^22-1034509294864058981085023270603830026423327450703467479894216670729726508295137433074776703715707851079504454343203907806500933427518600020803912290870823201006036937634388476368630601642792699392002030656764381835728322090450413433858327763983830074410401792 x^23+37538875608004301896183798420453696906822879159335832251290763559279870884714920418701927376664781518825083333093528570806440787171214924616100845062188200225263888596031677526660206087310554760293554414835647956131422466407939756102693429619819316267483922432 x^24-1066718079494739333596284097937492449546563079310476859234939838852074105977637965992430088091391427009466674865900162913382224901665722750042834633704441031635530082631744077070909024246396235048073115146618880417230655211168434508015774962131327020782994849792 x^25+23480647814483711777005434220085815074552935360069991084297502654977578448094655801846465545417230835556630915078987052616445467636301644465142962002907206067098712795716050618068749075917105356422255781432621095490617809605875358004054147804990298649431178739712 x^26-393915986891642363298510263670445963288762683207844396104049069452772055707059101997038771674131819898487673329216385489093291083771385667800567031741044620309199244625449760568409409688998593621901944812663195910421201239874988841964712100485916551164384803553280 x^27+4915003000079071133857608344736075417457337130910768698043246682986067832560492091811440527777672494203926253797046675181053534479162181643919054686807689806661639166291932541865172614371007135544491942357131495610499344563735544653643969859823928509304019151749120 x^28-43915483327309032316508244506713196308680777275320580609851439510586741026067006432055525747316009448680530455964021135039041515705562292207679796310307134640121135663383004242276533753172339202965242531255592912622490160994720128539700572881579618521991122180112384 x^29+263903983417715989937984896104510013908771304929542154421689441826065549033838013488278710296812590473535499758375559886392768079697986686153379360791198908445481621114792778559908241046225177256517821175702164530639642466408436972854148180434220411824527385303343104 x^30-948268119208765498966858691427941696706660432014606487637541331271815927163486893222049724409107486261010131516307103772743791649469878912967445864601010362492246243745307720223632161652161851632447415750504642061668134922960914836296784702814016477336769254141011968 x^31+1524307411995722575380934996670428710537817339549256899727383476361328976611532940321430596810311504804037382552628432694868152127396263371970621168709246920726181681678186641950136807646207344177850033058330590246617345554648137787398131199204207457762569572798431103 x^32

wayne

mathe的计算结果我也证实了,就是 最大值 是怎么得到的, 都枚举一遍吗.

1. f=#^2/(1+#^2)&;g=(2#)/(1-#^2)&;
   
2. n=5;eqs=Numerator[Together[2^Range[0,n].(1/NestList[g,x,n])]];ans=RootReduce[Solve[eqs==0,x,Reals]];
   
3. MaximalBy[Table[{x/.i,RootReduce[Times@@f/@NestList[g,x,n]/.i]},{i,ans}],Last]

复制代码

1. {2,-3+9 x^2-2 x^4}
   
2. {3,-4+55 x^2-129 x^4+49 x^6-3 x^8}
   
3. {4,-5+287 x^2-3769 x^4+15631 x^6-24819 x^8+15189 x^10-3531 x^12+253 x^14-4 x^16}
   
4. {5,-6+1383 x^2-83269 x^4+1936483 x^6-21641065 x^8+129870011 x^10-448777105 x^12+930130167 x^14-1180658289 x^16+923306757 x^18-441953695 x^20+126714521 x^22-20869723 x^24+1837593 x^26-77131 x^28+1229 x^30-5 x^32}
   
5. {6,-7+6375 x^2-1620373 x^4+173382107 x^6-9719028577 x^8+322364565735 x^10-6848540613413 x^12+98415788065883 x^14-994825867957133 x^16+7281860429763651 x^18-39454123980235081 x^20+160915415089864839 x^22-500420894103944213 x^24+1198134994878264851 x^26-2224168387548227465 x^28+3216624420038640279 x^30-3634194881023859879 x^32+3210557673455433333 x^34-2215696804481587135 x^36+1191172519726329681 x^38-496447188773327555 x^40+159265205234957589 x^42-38948059624730191 x^44+7167164993039601 x^46-975765693341183 x^48+96128567111969 x^50-6654895878563 x^52+311176203277 x^54-9298413447 x^56+163794737 x^58-1501891 x^60+5725 x^62-6 x^64}
   
6. {7,-8+28615 x^2-29588005 x^4+13439007851 x^6-3327450877513 x^8+507688468298975 x^10-51717902493870557 x^12+3720579567346040643 x^14-196973825622048321025 x^16+7919417003381731640303 x^18-247855930914186172102397 x^20+6159728224861226309612339 x^22-123553219464960043568475841 x^24+2027489338559208746983936999 x^26-27531160680221767549422595797 x^28+312361920296703709188178029163 x^30-2985813955859532539848679091769 x^32+24218413482842359984050585232179 x^34-167725742826035694179127644362089 x^36+997165283676694622247025980885415 x^38-5113169821301542676034237731985245 x^40+22706561049807189415201819819384555 x^42-87640656565022259970670913853903297 x^44+294921804884415929793708194669435679 x^46-867619536767913421680539321446894133 x^48+2236582515055856154841814772968629835 x^50-5062182826453013990541296832899373457 x^52+10076646439877283618032725956081916063 x^54-17665293419601460845211308840201892965 x^56+27304524305652589531574973468528296675 x^58-37241438255932885498912684159618091785 x^60+44849416119369374416999218770454130295 x^62-47706822413420645777953933323232777421 x^64+44827917334695480093583211458599744245 x^66-37205643604212411830002656640392502735 x^68+27264984539466392259479188542864617825 x^70-17630969452100403751068291078755645115 x^72+10051970290268415544946123997852991933 x^74-5047154494736334938511110677887773047 x^76+2228740727338714202235346329214334825 x^78-864092806842214659926307481663216483 x^80+293550701830728165526635398891062029 x^82-87179354852903843172443185177768247 x^84+22572276750289157901363274904038105 x^86-5079389881901937571475911222961795 x^88+989837360314875525490986896917365 x^90-166358823040450925780034799807039 x^92+23999959976875445309487515049729 x^94-2956036215641820230257607669419 x^96+308918236053839020323904463313 x^98-27195416508844045408549331427 x^100+2000098318210965958792042829 x^102-121699691772221809781054431 x^104+6056772622108176822213689 x^106-243219821947603049544747 x^108+7752695193325916868853 x^110-192272660967586001575 x^112+3618977992052550393 x^114-50082934615584507 x^116+488826141259925 x^118-3179262000663 x^120+12702592001 x^122-27523747 x^124+25949 x^126-7 x^128}

复制代码

mathe

记$f_n(x)=\prod_{k=0}^n\sin^2(2^kx)$, 于是我们有$f_n(\frac{\pi}3)=(\frac 3 4)^{n+1}$
另外题目条件已经给出$f_n(x)\le (\frac 3 4)^n$
由于$f_n(x)=\sin^2(x)f_{n-1}(2x)$, 所以我们马上得出
引论1：如果$|sin(x)|\lt\frac 3 4$,那么$f_n(x)$不可能取到这个函数的最大值。
     因为这时$f_n(x)=\sin^2(x)f_{n-1}(2x)\lt (\frac 3 4)^2(\frac 3 4)^{n-1}=(\frac 3 4)^{n+1}=f_n(\frac{\pi}3)$
同样我们还有
引论2：如果有$0\le k\le n$使得$|sin(2^kx)|\lt \frac{3\sqrt{3}}8$, 那么同样$f_n(x)$不可能取到这个函数的最大值。
因为这时$f_n(x)=f_{k-1}(x)\sin^2(2^k x) f_{n-k-1}(2^{k+1}x)\le (\frac 3 4)^{k-1} ( \frac{3\sqrt{3}}8)^2(\frac 3 4)^{n-k-1}=f_n(\frac{\pi}3)$。

引论1表示$f_n(x)$在$(0,\pi)$上最大值必然在子区间$(\frac{\pi}4,\frac{3\pi}4)$, 而且由于对称性，我们后面只需要考虑区间$(\frac{\pi}4,\frac{\pi}2)$上的最大值。
现在假设$f_n(x)$在$(\frac{\pi}4,\frac{\pi}2)$中最大值在$\frac{\pi}3+d$取到，其中$|d|<\frac{\pi}6$,那么如果$|d|\gt\frac{2\pi}{15\times 2^{n-2}}$
那么必然存在整数h使得$1\le h\le n-2$,使得$|2^{h-1} d|\le \frac{2\pi}15$但是$|2^h d| \gt \frac{2\pi} 15$,或者说$ \frac{2\pi}{15} \lt |2^h d| \le \frac{4\pi} 15$
于是$2^h  (\frac{\pi}3+d)=2^h \frac{\pi}3 + 2^h d$
于是如果$d<0$,我们得出$2^h\frac{\pi}3 - \frac{4\pi} {15} \le 2^h  (\frac{\pi}3+d)\lt 2^h \frac{\pi}3 - \frac{2\pi}{15}$, $2^{h+1} \frac{\pi}3 - \frac{8\pi} {15}  \le 2^{h+1}  (\frac{\pi}3+d)\lt 2^{h+1} \frac{\pi}3 - \frac{4\pi}{15}$,
   其中h或h+1必然有一个是偶数，于是对应数据的正弦值总是不超过$|sin(\frac{\pi}{5})|<\frac{3\sqrt{3}}8$
如果$d>0$,我们得出$2^h\frac{\pi}3 + \frac{2\pi}{15} \lt 2^h  (\frac{\pi}3+d)\le 2^h \frac{\pi}3 +  \frac{4\pi} {15}$,  $2^{h+1}\frac{\pi}3 + \frac{4\pi}{15} \lt 2^{h+1}  (\frac{\pi}3+d)\le 2^{h+1}\frac{\pi}3 +  \frac{8\pi}{15}$
   其中h或h+1必然有一个是奇数，于是对应数据的正弦值同样不超过$|sin(\frac{4\pi}{5})|<\frac{3\sqrt{3}}8$
于是根据引论2，我们得出$|d|\le \frac{2\pi}{15\times 2^{n-2}}$, 也就是最大值在区间$[\frac{\pi}3 - \frac{2\pi}{15\times 2^{n-2}}, \frac{\pi}3 + \frac{2\pi}{15\times 2^{n-2}}]$取到

数学星空

我们可以先利用数值计算方法得到：

\(n=1 ,x=0.9553166180,S(1,x)=0.5925925927\)

\(n=2, x=1.113871858, S(2,x)=0.4723877416\)

\(n=3, x=1.020037970 ,S(3,x)=0.3417146295\)

\(n=4, x=1.062156021,S(4,x)=0.2604733070\)

\(n=5,x=1.040080400,S(5,x)=0.1937018282\)

\(n=6,x=1.050844294,S(6,x)=0.1458794554\)

\(n=7,x=1.045396504,S(7,x)=0.1091804079\)

\(n=8,x=1.048103620,S(8,x)=0.08197066428\)

\(n=9,x=1.046745908,S(9,x)=0.06144587996\)

我们绘图可以得到

红色点为\(S(n,x)=(\frac{3}{4})^n\)的点列

绿色为数值计算的最大值

黑色点列为数值计算的最大值对应的零点

黑色直线为\(\frac{\pi}{3}=1.047197551\)






因此可以确信mathe的猜测：

为了得到更精确的估计：我可以设最大值对应零点为\(x_n=\frac{\pi}{3}(1+\frac{a}{n})\)

我们渐近分析得到：

\(\sin(2^kx)=\frac{(-1)^k\sqrt{3}}{2}-\frac{2^k\pi a}{6n}-\frac{(-1)^{k+1}4^k(\pi)^2 a^2}{36n^2}+\frac{8^k(\pi)^3a^3}{324n^3}+\dots\)

\(\frac{2^k}{\tan(2^kx)}=\frac{(-1)^k2^k\sqrt{3}}{3}-\frac{4^{k+1}\pi a}{9n}+\frac{(-1)^k2^{3k+2}\sqrt{3}(\pi)^2a^2}{81n^2}+\dots\)

\(\sum_{k=0}^n \frac{2^k}{\tan(2^kx)} =-\frac{(-2)^{n+1}\sqrt{3}}{9}+\frac{\sqrt{3}}{9}+\frac{4(1-4^{n+1})\pi a}{27n}+\dots\)

取最后一个式子的前三项可以得到：

\(-\frac{(-2)^{n+1}\sqrt{3}}{9}+\frac{\sqrt{3}}{9}+\frac{4(1-4^{n+1})\pi a}{27n}=0\)

解得 \(a=\frac{(-1)^n3\sqrt{3}n(2^{n+1}-(-1)^{n+1})}{4\pi(4^{n+1}-1)}\)

谁有兴趣检验一下计算结果

此思路应该可以得到\(S(n,x)\)的渐近表达式

mathe

设$S(n,x)$的最大值在$x=\frac{\pi}3+(-1)^n \frac{u_n\pi}{2^n}$时取到，6#已经证明了$|u_n|\lt \frac{8}{15}$
而由于对应极值条件为$\sum_{k=0}^n \frac{2^k}{\tan(2^k x)}=0$
于是上面方程可以改造为$\sum_{k=0}^n \frac{(-2)^{-k}}{\tan(\frac{\pi}3 + (-2)^{-k} u_n\pi)}=0$, 其中$|u_n|\lt \frac{8}{15}$
改造以后的方程的优点是容易看出随着n的增大，$u_n$后面的变化会很小。
比如
\(\begin{cases}u_0=\frac 1 6\\ u_1=0.058493218697273936837522225889097577985\\
u_2=0.084892363556764800725766656117986720776 \\
u_3 = 0.069161306041355184275941745556086179701 \\
u_4 = 0.076182860324977237909034452102919069964 \\
u_5 = 0.072494703301028546917166261219525917720 \\
u_6 = 0.074290831900898337020180493526800342018 \\
u_7 = 0.073381262086601776841853527751272320985 \\
u_8 = 0.073833112485382340041954139116781566971 \\
u_9 = 0.073606461052076017385708446065926551547 \\
u_{10} = 0.073719604294196180634409853383982713923
\end{cases}\)
另一方面，如果计算$u_n$的极限，注意到k充分大时$\frac{(-2)^{-k}}{\tan(\frac{\pi}3 + (-2)^{-k} u_n\pi)}$近似等于$\frac{(-2)^{-k}}{\tan(\frac{\pi}3 )}$, 于是$\sum_{k=n+1}^{\infty} \frac{(-2)^{-k}}{\tan(\frac{\pi}3 + (-2)^{-k} u_n\pi)}\approx -\frac{(-2)^{-n}}{3\tan(\pi/3)}$
我们可以利用方程
$\sum_{k=0}^n \frac{(-2)^{-k}}{\tan(\frac{\pi}3 + (-2)^{-k} u_n\pi)}=\frac{(-2)^{-n}}{3\tan(\pi/3)}$
来近似计算极限
这种方法分别计算n=9,10,11,12,13,得到极限在0.073681928, 0.073681869, 0.073681854, 0.0736818506, 0.0736818497附近

数学星空

现在我们利用渐近分析给出最终结果：

设\(x=\frac{\pi}{3}+\frac{u_1}{t}+\frac{u_2}{t^2}+\frac{u_3}{t^3}+\frac{u_4}{t^4}+\frac{u_5}{t^5}+\frac{u_6}{t^6}\)

其中\(t=(-2)^n\)

为了方便输入分析计算结果，我们简记\(2^k=a, (-1)^k=x,\sqrt{3}=y,(-1)^n=z\)

\(\frac{2^k}{\tan(2^kx)}=\frac{ayx}{3} - \frac{4a^2u_1}{3t}+\frac{a(60a^2u_1^2xy - 180au_2)}{135t^2}+ \frac{a(-120a^3u_1^3 + 120a^2u_1u_2xy - 180au_3)}{135t^3} +\frac{a(60a^4u_1^4xy - 360a^3u_1^2u_2 + 120a^2u_1u_3xy + 60a^2u_2^2xy - 180au_4)}{135t^4} + \frac{a(-104a^5u_1^5 + 240a^4u_1^3u_2xy - 360a^3u_1^2u_3 - 360a^3u_1u_2^2 + 120a^2u_1u_4xy + 120a^2u_2u_3xy - 180au_5)}{135t^5} + \frac{a(56a^6u_1^6xy - 520a^5u_1^4u_2 + 240a^4u_1^3u_3xy + 360a^4u_1^2u_2^2xy - 360a^3u_1^2u_4 - 720a^3u_1u_2u_3 - 120a^3u_2^3 + 120a^2u_1u_5xy + 120a^2u_2u_4xy + 60a^2u_3^2xy - 180au_6)}{135t^6}+\dots\)

\(\sum_{k=0}^n \frac{2^k}{\tan(2^kx)}=(-\frac{128}{135}u_1^3 - \frac{16}{9}u_1 + \frac{128}{297}yu_1^4 + \frac{32}{81}yu_1^2 + \frac{7168}{17415}yu_1^6 - \frac{6656}{8505}u_1^5 + \frac{2}{9}y)t + \frac{64yu_1u_2}{81} + \frac{512yu_1^3u_2}{297} - \frac{128u_1^2u_2}{45} - \frac{6656u_1^4u_2}{1701} - \frac{16u_2}{9}+ \frac{y}{9} +\frac{(-16/9u_3 + 4/9u_1 + 64/81yu_1u_3 + 512/297yu_1^3u_3 + 256/99yu_1^2u_2^2 + 32/81yu_2^2 - 128/45u_1u_2^2 - 128/45u_1^2u_3}{t} + \frac{-16/9u_4 - 128/135u_2^3 + 4/9u_2 - 256/45u_1u_2u_3 + 64/81yu_1u_4 + 64/81yu_2u_3 + 4/81yu_1^2 - 128/45u_1^2u_4}{t^2} + \frac{-16/9u_5 + 4/9u_3 + 8/81yu_1u_2 + 64/81yu_1u_5 + 64/81yu_2u_4 + 32/81yu_3^2 + 8/135u_1^3}{t^3} + \frac{-16/9u_6 + 4/9u_4 + 8/81yu_1u_3 + 4/297yu_1^4 + 4/81yu_2^2 + 8/45u_1^2u_2}{t^4} + \frac{4/9u_5 + 8/81yu_1u_4 + 16/297yu_1^3u_2 + 8/81yu_2u_3 + 104/8505u_1^5 + 8/45u_1^2u_3 + 8/45u_1u_2^2}{t^5} + \frac{4/9u_6 + 16/45u_1u_2u_3 + 8/135u_2^3 + 8/81yu_1u_5 + 8/81yu_2u_4 + 16/297yu_1^3u_3 + 8/99yu_1^2u_2^2 + 4/81yu_3^2 + 56/17415yu_1^6 + 104/1701u_1^4u_2 + 8/45u_1^2u_4}{t^6}+\dots\)

由此得到：

\(-\frac{128}{135}u_1^3 - \frac{16}{9}u_1 + \frac{128}{297}yu_1^4 + \frac{32}{81}yu_1^2 + \frac{7168}{17415}yu_1^6 - \frac{6656}{8505}u_1^5 + \frac{2}{9}y=0\)

\(\frac{64yu_1u_2}{81} + \frac{512yu_1^3u_2}{297} - \frac{128u_1^2u_2}{45} - \frac{6656u_1^4u_2}{1701} - \frac{16u_2}{9}+ \frac{y}{9}=0\)

\(-\frac{16}{9}u_3 + \frac{4}{9}u_1 + \frac{64}{81}yu_1u_3 + \frac{512}{297}yu_1^3u_3 +\frac{256}{99}yu_1^2u_2^2 +\frac{32}{81}yu_2^2 - \frac{128}{45}u_1u_2^2 - \frac{128}{45}u_1^2u_3=0\)

\(-\frac{16}{9}u_4 - \frac{128}{135}u_2^3 + \frac{4}{9}u_2 - \frac{256}{45}u_1u_2u_3 + \frac{64}{81}yu_1u_4 + \frac{64}{81}yu_2u_3 + \frac{4}{81}yu_1^2 - \frac{128}{45}u_1^2u_4=0\)

\(-\frac{16}{9}u_5 + \frac{4}{9}u_3 + \frac{8}{81}yu_1u_2 + \frac{64}{81}yu_1u_5 + \frac{64}{81}yu_2u_4 +\frac{32}{81}yu_3^2 + \frac{8}{135}u_1^3=0\)

\(-\frac{16}{9}u_6 +\frac{4}{9}u_4 + \frac{8}{81}yu_1u_3 + \frac{4}{297}yu_1^4 + \frac{4}{81}yu_2^2 + \frac{8}{45}u_1^2u_2=0\)

可解得：

\({u_1 = 0.2314912188, u_2 = 0.1212208330, u_3 = 0.06774522625, u_4 = 0.03544953868, u_5 = 0.03057260894, u_6 = 0.01176569735}\)

\(\sin(2^kx)=\frac{-xy}{2}- \frac{au_1}{2t} + \frac{1/4a^2xyu_1^2 - 1/2au_2}{t^2} + \frac{1/2a^2xyu_1u_2 - 1/2au_3 + 1/12a^3u_1^3}{t^3} + \frac{-1/48xya^4u_1^4 + 1/4a^2xyu_2^2 + 1/2xya^2u_1u_3 + 1/4a^3u_1^2u_2 - 1/2au_4}{t^4} + \frac{1/2xya^2u_2u_3 - 1/12xya^4u_2u_1^3 + 1/2xya^2u_1u_4 + 1/4a^3u_1^2u_3 - 1/2au_5 - 1/240a^5u_1^5 + 1/4a^3u_1u_2^2}{t^5} + \frac{1/1440xya^6u_1^6 - 1/8xya^4u_1^2u_2^2 + 1/4xya^2u_3^2 + 1/2xya^2u_2u_4 - 1/12xya^4u_1^3u_3 + 1/2xya^2u_1u_5 - 1/48a^5u_2u_1^4 + 1/4a^3u_1^2u_4 - 1/2au_6 + 1/12a^3u_2^3 + 1/2a^3u_1u_2u_3}{t^6}+\dots\)

\(\sum_{k=0}^n\ln(\sin(2^kx)=(n+1)\ln(\frac{-\sqrt{3}}{2})+\frac{128}{1485}yu_1^5 + \frac{2}{9}yu_1 + \frac{32}{243}yu_1^3 - \frac{3328}{25515}u_1^6 - \frac{32}{135}u_1^4 - \frac{8}{9}u_1^2 + \frac{1/9yu_1 + 128/297yu_1^4u_2 + 32/81yu_1^2u_2 - 16/9u_1u_2 - 128/135u_2u_1^3 + 2/9yu_2}{t} + \frac{1/9yu_2 + 32/81yu_2^2u_1 + 32/81yu_1^2u_3 - 16/9u_1u_3 - 128/135u_1^3u_3 - 64/45u_1^2u_2^2 + 2/9yu_3 - 8/9u_2^2 + 2/9u_1^2}{t^2} + \frac{4/243yu_1^3 + 1/9yu_3 + 4/9u_1u_2 + 64/81yu_1u_2u_3 + 32/81yu_1^2u_4 - 16/9u_1u_4 + 32/243yu_2^3 + 2/9yu_4 - 16/9u_2u_3}{t^3} + \frac{1/9yu_4 + 2/9u_2^2 + 4/81yu_1^2u_2 - 16/9u_1u_5 - 16/9u_2u_4 + 2/9yu_5 - 8/9u_3^2 + 2/135u_1^4 + 4/9u_1u_3}{t^4} + \frac{4/1485yu_1^5 + 1/9yu_5 + 4/81yu_1^2u_3 + 4/81yu_2^2u_1 + 2/9yu_6 + 8/135u_2u_1^3 + 4/9u_1u_4 + 4/9u_2u_3}{t^5} + \frac{8/81yu_1u_2u_3 + 4/243yu_2^3 + 1/9yu_6 + 2/9u_3^2 + 4/297yu_1^4u_2 + 4/81yu_1^2u_4 + 8/135u_1^3u_3 + 4/45u_1^2u_2^2 + 52/25515u_1^6 + 4/9u_1u_5 + 4/9u_2u_4}{t^6}+\dots\)

\(\prod_{k=0}^n\sin(2^kx)=(\frac{-\sqrt{3}}{2})^{n+1}\exp(v_0)(1 + \frac{v_1}{t} + \frac{v_1^2/2 + v_2}{t^2} + \frac{v_3 + v_1v_2 + 1/6v_1^3}{t^3} + \frac{v_4 + v_1v_3 + 1/2v_2v_1^2 + 1/2v_2^2 + 1/24v_1^4}{t^4} + \frac{v_5 + v_1v_4 + 1/2v_3v_1^2 + v_3v_2 + 1/2v_1v_2^2 + 1/6v_2v_1^3 + 1/120v_1^5}{t^5} + \frac{v_6 + v_1v_5 + 1/2v_4v_1^2 + v_4v_2 + 1/2v_3^2 + v_3v_1v_2 + 1/6v_3v_1^3 + 1/4v_2^2v_1^2 + 1/6v_2^3 + 1/24v_2v_1^4 + 1/720v_1^6}{t^6}+\dots)\)

\(\prod_{k=0}^n\sin(2^kx)=(\frac{-\sqrt{3}}{2})^{n+1}\exp(v_0)(1+\frac{w_1}{t}+\frac{w_2}{t^2}+\frac{w_3}{t^3}+\frac{w_4}{t^4}+\frac{w_5}{t^5}+\frac{w_6}{t^6}+\dots)\)

其中：

\(v_0 = \frac{128}{1485}yu_1^5 +\frac{2}{9}yu_1 + \frac{32}{243}yu_1^3 - \frac{3328}{25515}u_1^6 - \frac{32}{135}u_1^4 - \frac{8}{9}u_1^2\)

\(v_1 = \frac{1}{9}yu_1 + \frac{128}{297}yu_1^4u_2 + \frac{32}{81}yu_1^2u_2 - \frac{16}{9}u_1u_2 - \frac{128}{135}u_2u_1^3 + \frac{2}{9}yu_2\)

\(v_2 = \frac{1}{9}yu_2 + \frac{32}{81}yu_2^2u_1 + \frac{32}{81}yu_1^2u_3 - \frac{16}{9}u_1u_3 - \frac{128}{135}u_1^3u_3 - \frac{64}{45}u_1^2u_2^2 + \frac{2}{9}yu_3 - \frac{8}{9}u_2^2 + \frac{2}{9}u_1^2\)

\(v_3 = \frac{4}{243}yu_1^3 + \frac{1}{9}yu_3 + \frac{4}{9}u_1u_2 +\frac{64}{81}yu_1u_2u_3 +\frac{32}{81}yu_1^2u_4 -\frac{16}{9}u_1u_4 +\frac{32}{243}yu_2^3 +\frac{2}{9}yu_4 -\frac{16}{9}u_2u_3\)

\(v_4 = \frac{1}{9}yu_4 + \frac{2}{9}u_2^2 + \frac{4}{81}yu_1^2u_2 - \frac{16}{9}u_1u_5 - \frac{16}{9}u_2u_4 + \frac{2}{9}yu_5 - \frac{8}{9}u_3^2 +\frac{2}{135}u_1^4 + \frac{4}{9}u_1u_3\)

\(v_5 = \frac{4}{1485}yu_1^5 + \frac{1}{9}yu_5 +\frac{4}{81}yu_1^2u_3 + \frac{4}{81}yu_2^2u_1 + \frac{2}{9}yu_6 + \frac{8}{135}u_2u_1^3 + \frac{4}{9}u_1u_4 + \frac{4}{9}u_2u_3\)

\(v_6 = \frac{8}{81}yu_1u_2u_3 + \frac{4}{243}yu_2^3 + \frac{1}{9}yu_6 + \frac{2}{9}u_3^2 + \frac{4}{297}yu_1^4u_2 + \frac{4}{81}yu_1^2u_4 + \frac{8}{135}u_1^3u_3 + \frac{4}{45}u_1^2u_2^2 + \frac{52}{25515}u_1^6 +\frac{4}{9}u_1u_5 + \frac{4}{9}u_2u_4\)

\(w_1 = -\frac{1}{4455}y(-1920u_1^4u_2 + 1408u_1^3u_2y - 1760u_1^2u_2 + 2640u_1u_2y - 495u_1 - 990u_2)\)

\(w_2 = -16/81yu_1^2u_2 + 128/891u_1^5u_2 + 32/243u_2u_1^3 + 2/27u_1u_2 + 8192/29403u_1^8u_2^2 + 192512/200475u_2^2u_1^6 - 137216/120285yu_1^5u_2^2 - 16384/40095yu_1^7u_2^2 + 265472/120285u_1^4u_2^2 - 3328/3645yu_1^3u_2^2 + 512/1215u_1^2u_2^2 + 13/54u_1^2 - 22/27u_2^2 - 128/1215yu_1^4u_2 + 1/9yu_2 + 32/81yu_1^2u_3 - 16/9u_1u_3 - 128/135u_1^3u_3 + 2/9yu_3\)

\(w_3,w_4,w_5,w_6\)表达太长略

代入\(u_1,u_2,u_3,u_4,u_5,u_6\)值可以得到

\({v_0 = 0.0436950351879121, v_1 = 0.0446002718761726, v_2 = 0.0232659703554701, v_3 = 0.0146271591609330, v_4 = 0.0051224100527021, v_5 = 0.0184030658248678, v_6 = 0.00900622571847738, w_1 = 0.0446002718761720, w_2 = 0.0242605624811846, w_3 = 0.0156796141239876, w_4 = 0.00606874303617570, w_5 = 0.0189988064877544, w_6 = 0.0100760233605138}\)

最终可以得到：

\(x=\frac{\pi}{3} + \frac{0.2314912188(-1)^n}{2^n} + \frac{0.1212208330}{4^n}+\frac{ 0.06774522625(-1)^n}{8^n} + \frac{0.03544953868}{16^n}+\frac{0.03057260894(-1)^n}{32^n} + \frac{0.01176569735}{64^n}+\dots\)

\(S(n,x)=0.818491716899662(\frac{3}{4})^n(1 + \frac{0.08920054376(-1)^n}{2^n}+ \frac{0.05051030921}{4^n}+\frac{0.03352328361(-1)^n}{8^n} +\frac{ 0.01412469106}{16^n}+ \frac{0.03929974068(-1)^n}{32^n}+\frac{0.02238706313}{64^n}+\dots)\)

我们可得检验上面结果的精确度：

对于最大值零点x的检验：
[n = 1,数值计算 0 .9553166180,公式计算 0 .9547330377, 公式计算误差 -0.5835803e-3],
[n = 2, 数值计算1.113871858, 公式计算1.113876380, 公式计算误差0.4522e-5],
[n = 3, 数值计算1.020037970, 公式计算1.020030676, 公式计算误差-0.7294e-5],
[n = 4,数值计算 1.062156021, 公式计算1.062156381, 公式计算误差3.60*10^(-7)],
[n = 5, 数值计算1.040080400, 公式计算1.040079796, 公式计算误差-6.04*10^(-7)],
[n = 6, 数值计算1.050844294, 公式计算1.050844457,公式计算误差 1.63*10^(-7)],
[n = 7, 数值计算1.045396504, 公式计算1.045396392, 公式计算误差-1.12*10^(-7)],
[n = 8, 数值计算1.048103620, 公式计算1.048103667, 公式计算误差4.7*10^(-8)],
[n = 9, 数值计算1.046745908, 公式计算1.046745882, 公式计算误差-2.6*10^(-8)]

对于最大值S(n,x)的检验：
[n = 1, 数值计算 0.5925925927,公式计算0 .5916721327,公式计算误差 -0.9204600e-3],
[n = 2,数值计算 0 .4723877416, 公式计算0.4724087957, 公式计算误差0.210541e-4],
[n = 3, 数值计算 0.3417146295, 公式计算0.3417017786, 公式计算误差-0.128509e-4],
[n = 4,数值计算 0 .2604733070, 公式计算0.2604729770, 公式计算误差-3.300*10^(-7)],
[n = 5,数值计算 0 .1937018282, 公式计算0.1936998808, 公式计算误差-0.19474e-5],
[n = 6, 数值计算 0.1458794554, 公式计算0.1458787903, 公式计算误差-6.651*10^(-7)],
[n = 7, 数值计算 0.1091804079,公式计算0 .1091796529, 公式计算误差-7.550*10^(-7)],
[n = 8, 数值计算 0.8197066428e-1, 公式计算0.8197020681e-1, 公式计算误差-4.5747*10^(-7)],
[n = 9, 数值计算 0.6144587996e-1, 公式计算0.6144549873e-1, 公式计算误差-3.8123*10^(-7)]