myininaya
  • myininaya
Evaluate lim (1^(1/n)+2^(1/n)+...+2007^(1/n)-2006)^n n-infnity
Mathematics
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schrodinger
  • schrodinger
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anonymous
  • anonymous
1
myininaya
  • myininaya
\[\lim_{n \rightarrow \infty}( \sqrt[n]{1}+\sqrt[n]{2}+...+\sqrt[n]{2007}-2006)^n\]
myininaya
  • myininaya
no i wish it were that easy andras

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myininaya
  • myininaya
there you go polpak :)
myininaya
  • myininaya
this was a question i did in my senior seminar class. its been a while since i looked at it
anonymous
  • anonymous
I misunderstood it :-)
anonymous
  • anonymous
this is not diverging??
myininaya
  • myininaya
no it converges
myininaya
  • myininaya
i put it into maple and maple is still evaluating lol
myininaya
  • myininaya
wolfram alpha no result can be reached
anonymous
  • anonymous
but the sum is surely bigger than 1 and if u take the nth power of it than it goes to infinity
myininaya
  • myininaya
yes!
myininaya
  • myininaya
you will get a big freaking number
anonymous
  • anonymous
than what am I misunderstanding? :)
myininaya
  • myininaya
the number does exist though
anonymous
  • anonymous
really??? I dont believe you
anonymous
  • anonymous
if you take lim 1.001^n as n tends to infinity. this is diverging for sure
myininaya
  • myininaya
i will give you the answer it is 2007!
myininaya
  • myininaya
thats factorial not exclamation
anonymous
  • anonymous
2007? surprising :) but I start to understand why it converges
anonymous
  • anonymous
ok :)
myininaya
  • myininaya
nasty number
anonymous
  • anonymous
yeah, I cannot imagine such a big number
myininaya
  • myininaya
i have to leave to go to the hospital i will be back later have fun
anonymous
  • anonymous
bye
anonymous
  • anonymous
43046025187604931301073495306021606919706444365300504437700726938653476658540482895682023255731671008083253191134701531977327216262277734910301434073329657780176301162545225328061928920600009311905398639804126820396042450460066202697109907775276623968872853518354346016759166458978964531046772922079680334268349395400553907217843631670817062494745980159863699220055641618598527015369240922663513465449205721247877490866588987319843422033263068274836269183109480555389276190778038346967829969281182891390688896155908109741132615117268473730816500283359164264798601830326973866516776844628492169888293607239521260874661077748006501467554897266946929770167198202236168539094714436855125596965348850278428419408219121959065193321094260460012675282320096826009233318736492824372516701872916530191875274006751331906949469628695531029803522907150511237974426186839709893928853134757666263702066048826067502979089490242203757191931995294333171004695915352384474640644871822174241620572300156183558707078250717736961028156366959859921677655296330168740251591833351738043564377433284560158900338387796347112169428063480631538972184250753425370550017347776082214497306198063123186364841851403471408935730920459851486382788780998473732852005808994277131462002778650116911176327529356272603080102462324046604967312994263057368736624383919299458455699953933902352832256995941578134414743773382025255763658469026381866173288278447517255899658106552083051273359946737694651653754212906940384507355916630452196467129515987238021880606693829465264036899907307073437748560421341264116300230386978307277778110708808733561649284732562121195906834971751633631368677060936633119693654851223848615055082338939110035147213318136450234671212414029203616097230229945589841558040806845598058569001772383673629931716052342324341754195802753961257749189489037546944046112698324226187886768203898532477568362512929472764193781917650659658067589854223239462278187257059803948080557352654440583879422632096576531138731341885525844991796066351797621828730134725090944592790746734838522090756233976604544592665981142091966145661214078994176588123329399477109684347998836680683905444196796967903868455537458498586039586192714740410794873963813087940474907257329609557753515220359377337931228007810105449664070738053828013106422871266860404152807998523787092304682888264491432008554538300105320931903925794071312499192163207973053010023657157473954259657147335725251796830290146559452482096881004043699938653430179325321654494642446065202143276556976131126842569675581676839245919623228558247939229608790965820538400773470442008795867418950422510595413786972491033509552770587574679332977203196498537188583431130439956129081170636512351452788148050467473426861861103423636253521296569875953548170759645070796206682201768579959521236079495302391872041609223373410350948157764821209719094151213454012939429195884071063418582607938136555832992865478019751326416465650511040663018822058464400624606019400679681798288602527431886049034667225741015869049545764966156382561886172391332447947496644586519007980781117082080832021553420905696111049082685844767480840155238277177489208624726067851558869384426174647894231769867840866104268905410839849955133611554998133231555205769515722036852728601400019910791908069304128485748623142491012155998505368257654300095344895483194357030280508241289209799818268192667474541677114456558535733451965986362511131759065483735917201820897713058943975031526862863020430238708330397037864688223445598303312445818485142622513047532003550144078124831069189543160890852663608664354949670516059153261874952616358879968885489548737646622245041726345296829806734518932623185858874882546597909273541916936731698992217560436798479067580757234793907609041523948571053586384066337965273797847228760266551673618545481050809715004563792359718325107000073822179462358731590133936457251061372964290031222109669342316112331026823503833605565706502583832321627244774932094320883639781575056681178593042644061088809543333197948184330206660117244042299054620498800979659047820907097057396078391671915038011463035486300770242567590029510130760822673560320175464663608120374850039936697908449562312313298509745332308767805630772303120987386658714052978743556161615121774189061709715577455710161127856156620873018528092261285229010322165874979537115540211581972921040583292511405755187844636520805885692646019073556593713213213966061460742034317424610461043611879519537027747897267860975084431182515818098260712653832833939941816573348964437662757953006243481263581284433181109714025483815515348964930744099459782257197557903674741764225044035484415670530754237680782608356816059915546837787342603034767450416784886939340579307297735893890161000774744097698174116456513007650344673840749164573609437889867638252075539415406715721915296914731453346589790878646350370973001832566070316839486662669948238129199687090116537463734411569263617697006515197071129488492505811526009419472254401552612450606182107762414856252135950753135618729683159942855997294281110315350168393614610318114744974613446518541468264872922131535963094889967606125896648382597476027556006107102066564749867715245556119208056973521319244885469993060840556037400970969657521430843028156368845211529581718876773095256883200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Is not so terrible a number
anonymous
  • anonymous
hahahaha :)))) what did u use to get that?
anonymous
  • anonymous
python
anonymous
  • anonymous
:O
anonymous
  • anonymous
Here's an interesting question, How many 0's are on the end of 13000! (factorial)
anonymous
  • anonymous
Is there something special about 13000!?
anonymous
  • anonymous
Nope, just a big number with a lot of 0's on the end of it.
anonymous
  • anonymous
6501?
anonymous
  • anonymous
I guess next year at number theory they will teach us this. But if you wish u could explain how to do it.
anonymous
  • anonymous
About half that many.
anonymous
  • anonymous
3251?
anonymous
  • anonymous
3248
anonymous
  • anonymous
How to find it?
anonymous
  • anonymous
Well, factorials are the product of the sequence of numbers from 1 to the number in question.
anonymous
  • anonymous
Yeah.
anonymous
  • anonymous
So to find how many 0's will be on the end, you need to find how many factors of 10 are in your product.
anonymous
  • anonymous
And we know that 10 is the product of 5 and 2
anonymous
  • anonymous
Since every other number in the sequence will have at least one factor of 2, but only every 5th number will have a factor of 5 we can see that the limiting factor will be the 5's. (There will be at least one factor of 2 for each factor of 5 that can combine to make a factor of 10).
anonymous
  • anonymous
With me so far?
anonymous
  • anonymous
sure, this is clear
anonymous
  • anonymous
Yeah. Go on!
anonymous
  • anonymous
Ok, so we want to know how many factors of 5 are in the product 13000!, and that will be the same as the number of factors of 10, and the number of 0's at the end of our product.
anonymous
  • anonymous
To find how many factors, we can rely on the fact that every 5th number will have at least 1 factor of 5. And of those that have 1 factor of 5, every 5th one of them will have a second factor, and so on.
anonymous
  • anonymous
There are a few ways of determining this, but the one I like best is successive division.
anonymous
  • anonymous
Sounds nice but I dont know what is it
anonymous
  • anonymous
floor(13000/5) = 2600 floor(2600/5) = 520 floor(520/5) = 104 floor(104/5) = 20 floor(20/5) = 4 So there are 2600 + 520 + 104 + 20 + 4 = 3248 factors of 5 in 13000
anonymous
  • anonymous
err 13000!
anonymous
  • anonymous
I see, it isnt that hard after all
anonymous
  • anonymous
Not hard, just takes some thinking.
myininaya
  • myininaya
if you can read this and if you are interested
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myininaya
  • myininaya
oh hey polpak this wasn't the cool problem i was talking about but this problem is cool lol

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