Question

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

Answer 1

1

Answer 2

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

Answer 3

no i wish it were that easy andras

Answer 4

there you go polpak :)

Answer 5

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

Answer 6

I misunderstood it :-)

Answer 7

this is not diverging??

Answer 8

no it converges

Answer 9

i put it into maple and maple is still evaluating lol

Answer 10

wolfram alpha no result can be reached

Answer 11

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

Answer 12

yes!

Answer 13

you will get a big freaking number

Answer 14

than what am I misunderstanding? :)

Answer 15

the number does exist though

Answer 16

really??? I dont believe you

Answer 17

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

Answer 18

i will give you the answer
it is 2007!

Answer 19

thats factorial not exclamation

Answer 20

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

Answer 21

ok :)

Answer 22

nasty number

Answer 23

yeah, I cannot imagine such a big number

Answer 24

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

Answer 25

bye

Answer 26

43046025187604931301073495306021606919706444365300504437700726938653476658540482895682023255731671008083253191134701531977327216262277734910301434073329657780176301162545225328061928920600009311905398639804126820396042450460066202697109907775276623968872853518354346016759166458978964531046772922079680334268349395400553907217843631670817062494745980159863699220055641618598527015369240922663513465449205721247877490866588987319843422033263068274836269183109480555389276190778038346967829969281182891390688896155908109741132615117268473730816500283359164264798601830326973866516776844628492169888293607239521260874661077748006501467554897266946929770167198202236168539094714436855125596965348850278428419408219121959065193321094260460012675282320096826009233318736492824372516701872916530191875274006751331906949469628695531029803522907150511237974426186839709893928853134757666263702066048826067502979089490242203757191931995294333171004695915352384474640644871822174241620572300156183558707078250717736961028156366959859921677655296330168740251591833351738043564377433284560158900338387796347112169428063480631538972184250753425370550017347776082214497306198063123186364841851403471408935730920459851486382788780998473732852005808994277131462002778650116911176327529356272603080102462324046604967312994263057368736624383919299458455699953933902352832256995941578134414743773382025255763658469026381866173288278447517255899658106552083051273359946737694651653754212906940384507355916630452196467129515987238021880606693829465264036899907307073437748560421341264116300230386978307277778110708808733561649284732562121195906834971751633631368677060936633119693654851223848615055082338939110035147213318136450234671212414029203616097230229945589841558040806845598058569001772383673629931716052342324341754195802753961257749189489037546944046112698324226187886768203898532477568362512929472764193781917650659658067589854223239462278187257059803948080557352654440583879422632096576531138731341885525844991796066351797621828730134725090944592790746734838522090756233976604544592665981142091966145661214078994176588123329399477109684347998836680683905444196796967903868455537458498586039586192714740410794873963813087940474907257329609557753515220359377337931228007810105449664070738053828013106422871266860404152807998523787092304682888264491432008554538300105320931903925794071312499192163207973053010023657157473954259657147335725251796830290146559452482096881004043699938653430179325321654494642446065202143276556976131126842569675581676839245919623228558247939229608790965820538400773470442008795867418950422510595413786972491033509552770587574679332977203196498537188583431130439956129081170636512351452788148050467473426861861103423636253521296569875953548170759645070796206682201768579959521236079495302391872041609223373410350948157764821209719094151213454012939429195884071063418582607938136555832992865478019751326416465650511040663018822058464400624606019400679681798288602527431886049034667225741015869049545764966156382561886172391332447947496644586519007980781117082080832021553420905696111049082685844767480840155238277177489208624726067851558869384426174647894231769867840866104268905410839849955133611554998133231555205769515722036852728601400019910791908069304128485748623142491012155998505368257654300095344895483194357030280508241289209799818268192667474541677114456558535733451965986362511131759065483735917201820897713058943975031526862863020430238708330397037864688223445598303312445818485142622513047532003550144078124831069189543160890852663608664354949670516059153261874952616358879968885489548737646622245041726345296829806734518932623185858874882546597909273541916936731698992217560436798479067580757234793907609041523948571053586384066337965273797847228760266551673618545481050809715004563792359718325107000073822179462358731590133936457251061372964290031222109669342316112331026823503833605565706502583832321627244774932094320883639781575056681178593042644061088809543333197948184330206660117244042299054620498800979659047820907097057396078391671915038011463035486300770242567590029510130760822673560320175464663608120374850039936697908449562312313298509745332308767805630772303120987386658714052978743556161615121774189061709715577455710161127856156620873018528092261285229010322165874979537115540211581972921040583292511405755187844636520805885692646019073556593713213213966061460742034317424610461043611879519537027747897267860975084431182515818098260712653832833939941816573348964437662757953006243481263581284433181109714025483815515348964930744099459782257197557903674741764225044035484415670530754237680782608356816059915546837787342603034767450416784886939340579307297735893890161000774744097698174116456513007650344673840749164573609437889867638252075539415406715721915296914731453346589790878646350370973001832566070316839486662669948238129199687090116537463734411569263617697006515197071129488492505811526009419472254401552612450606182107762414856252135950753135618729683159942855997294281110315350168393614610318114744974613446518541468264872922131535963094889967606125896648382597476027556006107102066564749867715245556119208056973521319244885469993060840556037400970969657521430843028156368845211529581718876773095256883200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Is not so terrible a number

Answer 27

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

Answer 28

python

Answer 29

:O

Answer 30

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

Answer 31

Is there something special about 13000!?

Answer 32

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

Answer 33

6501?

Answer 34

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

Answer 35

About half that many.

Answer 36

3251?

Answer 37

3248

Answer 38

How to find it?

Answer 39

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

Answer 40

Yeah.

Answer 41

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.

Answer 42

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

Answer 43

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).

Answer 44

With me so far?

Answer 45

sure, this is clear

Answer 46

Yeah. Go on!

Answer 47

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.

Answer 48

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.

Answer 49

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

Answer 50

Sounds nice but I dont know what is it

Answer 51

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

Answer 52

err 13000!

Answer 53

I see, it isnt that hard after all

Answer 54

Not hard, just takes some thinking.

Answer 55

if you can read this and if you are interested

Answer 56

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