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myininaya

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    1

  2. myininaya
    • 5 years ago
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    \[\lim_{n \rightarrow \infty}( \sqrt[n]{1}+\sqrt[n]{2}+...+\sqrt[n]{2007}-2006)^n\]

  3. myininaya
    • 5 years ago
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    no i wish it were that easy andras

  4. myininaya
    • 5 years ago
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    there you go polpak :)

  5. myininaya
    • 5 years ago
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    this was a question i did in my senior seminar class. its been a while since i looked at it

  6. anonymous
    • 5 years ago
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    I misunderstood it :-)

  7. anonymous
    • 5 years ago
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    this is not diverging??

  8. myininaya
    • 5 years ago
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    no it converges

  9. myininaya
    • 5 years ago
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    i put it into maple and maple is still evaluating lol

  10. myininaya
    • 5 years ago
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    wolfram alpha no result can be reached

  11. anonymous
    • 5 years ago
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    but the sum is surely bigger than 1 and if u take the nth power of it than it goes to infinity

  12. myininaya
    • 5 years ago
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    yes!

  13. myininaya
    • 5 years ago
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    you will get a big freaking number

  14. anonymous
    • 5 years ago
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    than what am I misunderstanding? :)

  15. myininaya
    • 5 years ago
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    the number does exist though

  16. anonymous
    • 5 years ago
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    really??? I dont believe you

  17. anonymous
    • 5 years ago
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    if you take lim 1.001^n as n tends to infinity. this is diverging for sure

  18. myininaya
    • 5 years ago
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    i will give you the answer it is 2007!

  19. myininaya
    • 5 years ago
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    thats factorial not exclamation

  20. anonymous
    • 5 years ago
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    2007? surprising :) but I start to understand why it converges

  21. anonymous
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    ok :)

  22. myininaya
    • 5 years ago
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    nasty number

  23. anonymous
    • 5 years ago
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    yeah, I cannot imagine such a big number

  24. myininaya
    • 5 years ago
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    i have to leave to go to the hospital i will be back later have fun

  25. anonymous
    • 5 years ago
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    bye

  26. anonymous
    • 5 years ago
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    43046025187604931301073495306021606919706444365300504437700726938653476658540482895682023255731671008083253191134701531977327216262277734910301434073329657780176301162545225328061928920600009311905398639804126820396042450460066202697109907775276623968872853518354346016759166458978964531046772922079680334268349395400553907217843631670817062494745980159863699220055641618598527015369240922663513465449205721247877490866588987319843422033263068274836269183109480555389276190778038346967829969281182891390688896155908109741132615117268473730816500283359164264798601830326973866516776844628492169888293607239521260874661077748006501467554897266946929770167198202236168539094714436855125596965348850278428419408219121959065193321094260460012675282320096826009233318736492824372516701872916530191875274006751331906949469628695531029803522907150511237974426186839709893928853134757666263702066048826067502979089490242203757191931995294333171004695915352384474640644871822174241620572300156183558707078250717736961028156366959859921677655296330168740251591833351738043564377433284560158900338387796347112169428063480631538972184250753425370550017347776082214497306198063123186364841851403471408935730920459851486382788780998473732852005808994277131462002778650116911176327529356272603080102462324046604967312994263057368736624383919299458455699953933902352832256995941578134414743773382025255763658469026381866173288278447517255899658106552083051273359946737694651653754212906940384507355916630452196467129515987238021880606693829465264036899907307073437748560421341264116300230386978307277778110708808733561649284732562121195906834971751633631368677060936633119693654851223848615055082338939110035147213318136450234671212414029203616097230229945589841558040806845598058569001772383673629931716052342324341754195802753961257749189489037546944046112698324226187886768203898532477568362512929472764193781917650659658067589854223239462278187257059803948080557352654440583879422632096576531138731341885525844991796066351797621828730134725090944592790746734838522090756233976604544592665981142091966145661214078994176588123329399477109684347998836680683905444196796967903868455537458498586039586192714740410794873963813087940474907257329609557753515220359377337931228007810105449664070738053828013106422871266860404152807998523787092304682888264491432008554538300105320931903925794071312499192163207973053010023657157473954259657147335725251796830290146559452482096881004043699938653430179325321654494642446065202143276556976131126842569675581676839245919623228558247939229608790965820538400773470442008795867418950422510595413786972491033509552770587574679332977203196498537188583431130439956129081170636512351452788148050467473426861861103423636253521296569875953548170759645070796206682201768579959521236079495302391872041609223373410350948157764821209719094151213454012939429195884071063418582607938136555832992865478019751326416465650511040663018822058464400624606019400679681798288602527431886049034667225741015869049545764966156382561886172391332447947496644586519007980781117082080832021553420905696111049082685844767480840155238277177489208624726067851558869384426174647894231769867840866104268905410839849955133611554998133231555205769515722036852728601400019910791908069304128485748623142491012155998505368257654300095344895483194357030280508241289209799818268192667474541677114456558535733451965986362511131759065483735917201820897713058943975031526862863020430238708330397037864688223445598303312445818485142622513047532003550144078124831069189543160890852663608664354949670516059153261874952616358879968885489548737646622245041726345296829806734518932623185858874882546597909273541916936731698992217560436798479067580757234793907609041523948571053586384066337965273797847228760266551673618545481050809715004563792359718325107000073822179462358731590133936457251061372964290031222109669342316112331026823503833605565706502583832321627244774932094320883639781575056681178593042644061088809543333197948184330206660117244042299054620498800979659047820907097057396078391671915038011463035486300770242567590029510130760822673560320175464663608120374850039936697908449562312313298509745332308767805630772303120987386658714052978743556161615121774189061709715577455710161127856156620873018528092261285229010322165874979537115540211581972921040583292511405755187844636520805885692646019073556593713213213966061460742034317424610461043611879519537027747897267860975084431182515818098260712653832833939941816573348964437662757953006243481263581284433181109714025483815515348964930744099459782257197557903674741764225044035484415670530754237680782608356816059915546837787342603034767450416784886939340579307297735893890161000774744097698174116456513007650344673840749164573609437889867638252075539415406715721915296914731453346589790878646350370973001832566070316839486662669948238129199687090116537463734411569263617697006515197071129488492505811526009419472254401552612450606182107762414856252135950753135618729683159942855997294281110315350168393614610318114744974613446518541468264872922131535963094889967606125896648382597476027556006107102066564749867715245556119208056973521319244885469993060840556037400970969657521430843028156368845211529581718876773095256883200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Is not so terrible a number

  27. anonymous
    • 5 years ago
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    hahahaha :)))) what did u use to get that?

  28. anonymous
    • 5 years ago
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    python

  29. anonymous
    • 5 years ago
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    :O

  30. anonymous
    • 5 years ago
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    Here's an interesting question, How many 0's are on the end of 13000! (factorial)

  31. anonymous
    • 5 years ago
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    Is there something special about 13000!?

  32. anonymous
    • 5 years ago
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    Nope, just a big number with a lot of 0's on the end of it.

  33. anonymous
    • 5 years ago
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    6501?

  34. anonymous
    • 5 years ago
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    I guess next year at number theory they will teach us this. But if you wish u could explain how to do it.

  35. anonymous
    • 5 years ago
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    About half that many.

  36. anonymous
    • 5 years ago
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    3251?

  37. anonymous
    • 5 years ago
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    3248

  38. anonymous
    • 5 years ago
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    How to find it?

  39. anonymous
    • 5 years ago
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    Well, factorials are the product of the sequence of numbers from 1 to the number in question.

  40. anonymous
    • 5 years ago
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    Yeah.

  41. anonymous
    • 5 years ago
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    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.

  42. anonymous
    • 5 years ago
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    And we know that 10 is the product of 5 and 2

  43. anonymous
    • 5 years ago
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    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).

  44. anonymous
    • 5 years ago
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    With me so far?

  45. anonymous
    • 5 years ago
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    sure, this is clear

  46. anonymous
    • 5 years ago
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    Yeah. Go on!

  47. anonymous
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    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.

  48. anonymous
    • 5 years ago
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    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.

  49. anonymous
    • 5 years ago
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    There are a few ways of determining this, but the one I like best is successive division.

  50. anonymous
    • 5 years ago
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    Sounds nice but I dont know what is it

  51. anonymous
    • 5 years ago
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    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

  52. anonymous
    • 5 years ago
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    err 13000!

  53. anonymous
    • 5 years ago
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    I see, it isnt that hard after all

  54. anonymous
    • 5 years ago
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    Not hard, just takes some thinking.

  55. myininaya
    • 5 years ago
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    if you can read this and if you are interested

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  56. myininaya
    • 5 years ago
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    oh hey polpak this wasn't the cool problem i was talking about but this problem is cool lol

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