Evaluate lim (1^(1/n)+2^(1/n)+...+2007^(1/n)-2006)^n n-infnity

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Evaluate lim (1^(1/n)+2^(1/n)+...+2007^(1/n)-2006)^n n-infnity

Mathematics
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1
\[\lim_{n \rightarrow \infty}( \sqrt[n]{1}+\sqrt[n]{2}+...+\sqrt[n]{2007}-2006)^n\]
no i wish it were that easy andras

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

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