## myininaya 5 years ago Evaluate lim (1^(1/n)+2^(1/n)+...+2007^(1/n)-2006)^n n-infnity

1. anonymous

1

2. myininaya

$\lim_{n \rightarrow \infty}( \sqrt[n]{1}+\sqrt[n]{2}+...+\sqrt[n]{2007}-2006)^n$

3. myininaya

no i wish it were that easy andras

4. myininaya

there you go polpak :)

5. myininaya

this was a question i did in my senior seminar class. its been a while since i looked at it

6. anonymous

I misunderstood it :-)

7. anonymous

this is not diverging??

8. myininaya

no it converges

9. myininaya

i put it into maple and maple is still evaluating lol

10. myininaya

wolfram alpha no result can be reached

11. anonymous

but the sum is surely bigger than 1 and if u take the nth power of it than it goes to infinity

12. myininaya

yes!

13. myininaya

you will get a big freaking number

14. anonymous

than what am I misunderstanding? :)

15. myininaya

the number does exist though

16. anonymous

really??? I dont believe you

17. anonymous

if you take lim 1.001^n as n tends to infinity. this is diverging for sure

18. myininaya

i will give you the answer it is 2007!

19. myininaya

thats factorial not exclamation

20. anonymous

2007? surprising :) but I start to understand why it converges

21. anonymous

ok :)

22. myininaya

nasty number

23. anonymous

yeah, I cannot imagine such a big number

24. myininaya

i have to leave to go to the hospital i will be back later have fun

25. anonymous

bye

26. anonymous

43046025187604931301073495306021606919706444365300504437700726938653476658540482895682023255731671008083253191134701531977327216262277734910301434073329657780176301162545225328061928920600009311905398639804126820396042450460066202697109907775276623968872853518354346016759166458978964531046772922079680334268349395400553907217843631670817062494745980159863699220055641618598527015369240922663513465449205721247877490866588987319843422033263068274836269183109480555389276190778038346967829969281182891390688896155908109741132615117268473730816500283359164264798601830326973866516776844628492169888293607239521260874661077748006501467554897266946929770167198202236168539094714436855125596965348850278428419408219121959065193321094260460012675282320096826009233318736492824372516701872916530191875274006751331906949469628695531029803522907150511237974426186839709893928853134757666263702066048826067502979089490242203757191931995294333171004695915352384474640644871822174241620572300156183558707078250717736961028156366959859921677655296330168740251591833351738043564377433284560158900338387796347112169428063480631538972184250753425370550017347776082214497306198063123186364841851403471408935730920459851486382788780998473732852005808994277131462002778650116911176327529356272603080102462324046604967312994263057368736624383919299458455699953933902352832256995941578134414743773382025255763658469026381866173288278447517255899658106552083051273359946737694651653754212906940384507355916630452196467129515987238021880606693829465264036899907307073437748560421341264116300230386978307277778110708808733561649284732562121195906834971751633631368677060936633119693654851223848615055082338939110035147213318136450234671212414029203616097230229945589841558040806845598058569001772383673629931716052342324341754195802753961257749189489037546944046112698324226187886768203898532477568362512929472764193781917650659658067589854223239462278187257059803948080557352654440583879422632096576531138731341885525844991796066351797621828730134725090944592790746734838522090756233976604544592665981142091966145661214078994176588123329399477109684347998836680683905444196796967903868455537458498586039586192714740410794873963813087940474907257329609557753515220359377337931228007810105449664070738053828013106422871266860404152807998523787092304682888264491432008554538300105320931903925794071312499192163207973053010023657157473954259657147335725251796830290146559452482096881004043699938653430179325321654494642446065202143276556976131126842569675581676839245919623228558247939229608790965820538400773470442008795867418950422510595413786972491033509552770587574679332977203196498537188583431130439956129081170636512351452788148050467473426861861103423636253521296569875953548170759645070796206682201768579959521236079495302391872041609223373410350948157764821209719094151213454012939429195884071063418582607938136555832992865478019751326416465650511040663018822058464400624606019400679681798288602527431886049034667225741015869049545764966156382561886172391332447947496644586519007980781117082080832021553420905696111049082685844767480840155238277177489208624726067851558869384426174647894231769867840866104268905410839849955133611554998133231555205769515722036852728601400019910791908069304128485748623142491012155998505368257654300095344895483194357030280508241289209799818268192667474541677114456558535733451965986362511131759065483735917201820897713058943975031526862863020430238708330397037864688223445598303312445818485142622513047532003550144078124831069189543160890852663608664354949670516059153261874952616358879968885489548737646622245041726345296829806734518932623185858874882546597909273541916936731698992217560436798479067580757234793907609041523948571053586384066337965273797847228760266551673618545481050809715004563792359718325107000073822179462358731590133936457251061372964290031222109669342316112331026823503833605565706502583832321627244774932094320883639781575056681178593042644061088809543333197948184330206660117244042299054620498800979659047820907097057396078391671915038011463035486300770242567590029510130760822673560320175464663608120374850039936697908449562312313298509745332308767805630772303120987386658714052978743556161615121774189061709715577455710161127856156620873018528092261285229010322165874979537115540211581972921040583292511405755187844636520805885692646019073556593713213213966061460742034317424610461043611879519537027747897267860975084431182515818098260712653832833939941816573348964437662757953006243481263581284433181109714025483815515348964930744099459782257197557903674741764225044035484415670530754237680782608356816059915546837787342603034767450416784886939340579307297735893890161000774744097698174116456513007650344673840749164573609437889867638252075539415406715721915296914731453346589790878646350370973001832566070316839486662669948238129199687090116537463734411569263617697006515197071129488492505811526009419472254401552612450606182107762414856252135950753135618729683159942855997294281110315350168393614610318114744974613446518541468264872922131535963094889967606125896648382597476027556006107102066564749867715245556119208056973521319244885469993060840556037400970969657521430843028156368845211529581718876773095256883200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Is not so terrible a number

27. anonymous

hahahaha :)))) what did u use to get that?

28. anonymous

python

29. anonymous

:O

30. anonymous

Here's an interesting question, How many 0's are on the end of 13000! (factorial)

31. anonymous

Is there something special about 13000!?

32. anonymous

Nope, just a big number with a lot of 0's on the end of it.

33. anonymous

6501?

34. anonymous

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

36. anonymous

3251?

37. anonymous

3248

38. anonymous

How to find it?

39. anonymous

Well, factorials are the product of the sequence of numbers from 1 to the number in question.

40. anonymous

Yeah.

41. 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.

42. anonymous

And we know that 10 is the product of 5 and 2

43. 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).

44. anonymous

With me so far?

45. anonymous

sure, this is clear

46. anonymous

Yeah. Go on!

47. 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.

48. 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.

49. anonymous

There are a few ways of determining this, but the one I like best is successive division.

50. anonymous

Sounds nice but I dont know what is it

51. 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

52. anonymous

err 13000!

53. anonymous

I see, it isnt that hard after all

54. anonymous

Not hard, just takes some thinking.

55. myininaya

if you can read this and if you are interested

56. myininaya

oh hey polpak this wasn't the cool problem i was talking about but this problem is cool lol