watch casio new york

well i'm not sure if it's just me butanytime i hear the terminology improper integrals i have the same reactionevening even twenty years after act first heard the term interest what do you mean after we spend all this time trying todo the integrals right now we're trying to do a long time yet something in the work is just not methere but what improper integrals are ants are they are a vehicle

that allows us to you capture here yes dot otherwise we couldnot capture with three months sounds mehendi areas that have pieces accordingto be longer infinity tall so in this first presentation onimproper integrals what we'll talk about infinite intervals people's including which we could look at you with threemonths sums because i lost rectangle would always have in three weeks anti would not lead to a decent area

we will then separate presentation alsotalk about improper integrals crossing garrity sowe have one place what we have infinite hijack for thefunction went over x near zero but for this one we're going to talk about infiniteintervals and thought as far as i can see even though that's of coursecertainly still just my opinion improper integrals over into people'sare more frequent and probably more

important then improper integrals acrosssingularities even though those also happier applications so what's going on with these kinds of integrals let's just look an example in if we havea space probe that's is supposed to escape the gravitational field of theearth well then the amount of energy has atleast the integral over the force against

though this was a set the force travelsalong its way in fact would have to be because weweren't completely escape the gravitational fields the earth it wouldhave to be a vehicle that goes from the base of the earth all the way to infinity because eventhough we know that space is not infinite at least according tolatest observations of models would want to technically be able to getarbor charlie far away from the earth summit we have that we won't put want to

lengthy and it will go to infinity in that of course we integrate thegravitational force against the middle status that we make along the way soit's even for a larger infiniti of the gravitational force begins d_r_ or katie well similarity in probability theory and we'll talk about that and moredetail certainly never with the section but the probabilities and outcome liesin a certain in trouble is given by

the probability that the outcome isbetween a and b_ is the end of the for me to be over what is called aprobability density function will deal with that later uh... begins d_a_x_ if that would mean but the privileges and outcome is largerthan a number eight advocate probabilities that the outcome isbetween any infinity and that would have to be the end of mayto infinity just as previously placebo

from ada di insult that means that the first thing we haveto do is we have to four movie makes himself these integrals now thatwe have seen that apparently there are situations but we literallywant to integrate all the way to infinity okay so the definition is fairly simplewe work with the continuous function on atemple eight to infinity

if the illegal for eighteen years of expx exists with pre-teen in any infinity not aid be and we're back okay so we're saying that this intervalexists for every t between eighty and infinity uh... indeterminacy go cincinnati at theend of may two t ever sixty x exists within a defined the improper integralsthere for me to infinity

to be the integral from a definitegeographic sticks to be the business team goes to infinity of the immigrantsfrom a_t_t_ ethics d_x_ uh... in this case who will say that the improper integrals convert justlook at the property to quit improper enquiries convergence and if they did it does not exist whalesin the improper integrals it's called diverge

improper integrals rebels from a kittenconnected to any other point similarly it just makes you go to minus infinityright and of course we would put tea on thebottom sample fifty eight intellect eagle minus infinity uh... and finally if interested fine forall real numbers we can define the n_ double from minus infinity to infinity as asixty x two beating the from eight a m x the x plus the imprimatur infinity

ever sixty-eight sweetest split upsomewhere into the sunset can be proved that wheneverything is existing when everything existed works propertiesin the sums always back up to the same thingindependent authorities so trap of course on the website both in a gross on the right side exist individualization look here's where itgets a little bit rough because if we're looking at a typical x_y_coordinate system in eight functions as

defined for ages infinity well then the improper in april is the erie underneath the function from eachyear infinity into the infinity parties would be really have a hard time seeing because well that that just keeps goingright now the first thing we should workout is to show ourselvesthat is with infinitely still keen defying solet's take a look

at the improper at all from one to infinity of eternity excludeover to d_x_ well that a liberal yes well what do we do we take dat derivative right so that's a quicksubstitution uwe equals negative x over too d_u_ d_x_ is negative one-half into india to sniff too

do you live so when you were called outto ultimately end up with it within t derivative negative too you to the negative x_ hats and uh... if you want to double checkagain you could just take the derivative of this you end up with the integrity in we have to take that in theboundaries one 'cause that's where it starts team going to infinity

what do we help we know that yes the upper bound goes to infinity itthe exponential goes to zero as we end up with zero at the upper bound we met with minus negative to each of the native one-halfadjustment in the one at the lower bound and so that's to into the negative one-half which in particular is fine acts proximately one point to hunt freeone okay

uh... so basically we have any function best decays rather rapidly like anexponential function then the area under the function from an overnight won allthe way to infinity fine dot however the picture that it doesn't show was lyingfrighteningly we don't know how fast is competent to give you any at least i candevelop feeding four from the picture whether the seriousgoing to be fine alright on the other hand of course

improper integrals king also go lol wonder where x is a function that alsogoes to zero eight six goes to infinity including degraded over the same interval from one to infinityrecord to see that this one doesn't work out wellthat's eating to grow from one to infinity one for x the x the eightyderivative off the logarithmic scale in t if you take that in the boundaries fromone

to infinity well and that's infinity because thelogarithm is team goes to infinity goes to infinity also infinity minus zerothen of course is still infinity so this improper integrals i've purchased and if we want to look at that buthere's one over x here is the year yet yes admitted atlooks larger then the areas that we had under the exponential out you can't really conclude much fromthe finite sitting here so basically what you

really want to keep in mind is that thisis an infinite area and there is as demands all committee wewill not do people really williams the pictures visualize what's going on but thevisualization realty essex act as it was for finite uh... intervals of a finite intervals in okay cell

here's anne really neat uh... physics problems are simply compute the escape velocity from viewers willwhat does that mean that means we want to look at how much what thetotal energy has to get a part of mass in all the way from the radius of theearth to arbitrate distances to the technique from the orissa technicallyall the way to infinity we can compute the synergy

with exactly do you go from one toinfinite from larger infinity off the gravitational force times them distance is that the object travels ins then we're going to turn tax energy into kinetic energy which we can do with canadian inclinedin the square okay so

first of all we compute the integrity ofhere well g_e_ lower case in the capital m areconstants the integrity of over one of our oursquares native one of our aren't in then we go from the race of the earth all the way to infinity wrote themarking it doesn't work too well and well that means that the operacompany get something that i think that is so we get zero at the upper bound at the doorbound we get are in the minus minus gives usto us

so the energies g in capital m times aren't the radius of the earth is about six point three seven times tento the six meters yeah that's right it's about six thousand some connotes uh... as a mass of the earth isapproximately five point nine eight times ten to twenty four kilograms no those are not numbers thati expect you to know the numbers are

looked up just like you would the gravity of constant is six point sixseven plunged into the native eleven eaton has meters gratified by it square kilograms in so that means that in g_m_ over our is approximately the mass of the object ties all these numbers that are constantsalt nature ins if u work out that computations andthat's the last time six point two six times down to the sevens

newton meters per kilogram because see one of the kilograms and one of themeters cancel alright no problem in uh... that also makes sense becausethis is work for mass times mass which ultimatelygive us work or energy in sooner if we take that energyinsanity put to a kinetic energy with that said one-half envy square equal toingi and wilbur are which would give us that's the

velocity of other masarsky insulin so weend up with a philosophy beat the spirit of two times g_m_ over our if that would be the square root off two times the six point two seven timestend to that seven kilogram transmitter presents for attimes neater divided by kilograms and you canremember his kilogram times meter per second square

and we can see the kilograms cancel you may end up with square meters oversquare sickness in the spirit gives as meters per second so the sanity checkhere is yes this tuesday valid uh... a developed but also the unit everything that was approximately eleventhousand meters per second or about eleven condone occurs per second which if you want to lookthat up in

tables a reference works that is thatboth the escape velocity from the or if you might get slightlydifferent digits depending on how on how far people have drowned at all theseconcepts of nature and what need you great use for the earth andshows but that's about what the number is there and if you wantto turned that into you different units uh... and i think a dollars per secondsays about forty thousand

and some aka longer sprout work and that would be about twenty fivethousand and some miles per hour so it's it is rather fast now the neat thing of course the is thisis all not just theoretical uh... mumble jumble it's cake philosophy can be achieved forexample if escape velocity couldn't be achieved uh... jane this picture could have neverbeen taken to the picture

taken by that cassini spacecraft which i got fromeight which site that is still affiliated withan acess so with some that because there was no explicitcopyright notice this detective says a public domain image made available by nasa in cassini and i am certainly always very grateful

for this generous policy what they always say as it should beexplicitly said that you know and you think materials that they made available topeople this should not in anyway constitutive claymoreimplications at nasa endorses what i a m doing here i hope they do but i or i hope theywould

but um... i i cannot claim that uh... the unit itself one instance this is saturn but it looks kind of weird because well there's a light source behind satire we can't just put a really bigflashlight behind saturn and take a picture like that's what actuallyhappened here is the cassini spacecraft was

behind saccharine scene from the sun and so this is actually saturday keatingviewers it is just a a beautiful picture enrich somewhere around here i don't think i think this thing here that's for it's and this one here of this one here which was quite amazing the horsebecause mrs in hand

relation to what we're doing here if it was not part of the touristy the gravitational acceleration off the earth we would have never been able to move aspacecraft all the way out to saturn because the energies uh... would just be offering this itwould be required to do that sold now we're coming to the more technicalparts uh... improper immigration over infiniteintervals

and uh... you can see you were going tohave a convergence test america estimates absolute convergence and atthe end also have both uh... continuous factorial function the galfunction but um... so far it's just beentechnique but there are quite a few situations in which we cannot compute the integral for example for functions such as that standard normal distribution those things don't happen i think hedrew the picture was formed

and for these functions just provingthat the improper in the book converges it is finite is a good first step because then we cando in america lessons and yes after that we can to numericalestimates gabycs approximate value the idea for that for the most part forus will be the cooper computers and test which says the following let's take a look at function f from major infinity anotherfunction gene from a to infinity

but if the improper integrals ifconverges to f is always about g in the area under f is greater than thearea of the jesus the area under f is fine at the area under g must be finite uh... or conversely or equivalent reaction it uh... if the area under jeannie'sinfinite and it is always a dodgy democraticinfinity area inside the area under a different that means that thearea under f would have to be infinite so that's exactly what the fear of saysthe computers and just for improper

integrals over into intervals we take two continuous functions of theact this way rippled genius gregory wealth and zero forex gregory okay if the improv remote if converges to nvehicle over of g convert just so if the larger area is fine actor in a smallerarea must be fun activity equivalency eve the interloper g died purchased indianover f diverges too

beacause if the smaller areas infinite within the areas that contains a smallerarea has no choice but to be infinite also if we want to really visualize again here's thepicture we have to be very careful to not applied to compare isn't instrong because the integra from agency heads north everpicks d_x_ diverges that with me but the bigger area asking for that

that doesn't say anything about the areaunder g because any finite area is smaller or any areas so resonateswith me i can put an intermediary insight into the area accomplished find that areas into into similarity the compares the test doesnot apply winding improper envelope g cold burgess and that is because if the smaller area is fine at thelarger area

could be fined adoration right herethat's larger they couldn't find any re cindi finite infinite let's look at an example of the board todetermine if the integral from one to infinity eternity xc where there weretwo d_x_ corn for just or divergence this is another situation in which wehave two income for inequalities and

well what do we have here into negative excoriate over to you that's an explanation that altered it ought to d_k_ fairly quickly but really i would much rather comparedthis to something that you may have ex-lover true for which i could computethe integral and so the idea here is that x squared forex rhetoric with andwhat movie gregory polemics

well that means that make it a fakesquare dismissing recall the negative things that means that may get fixed great overtwo is this a recall the negative x overkill and hence ito made a fixed-rate over to assist inarticles and features a native x over so this is the derivation basically as i mentioned and want to compare this to you today tofix over too

but hiney too i need to correctly handle theinequalities in order to do that cute typically start out with any validating quality then he was for the loss redeemingqualities to build up the expression that you want to have we will see this and give these guysit's this is another spit in the direction of inequalities

that may be consider challenging becauseit is we haven't worked with inequalities very much sold for our model monica careers but that just means we have to takes an extra care when we have shown however is that youto me expiry of the two assessing accordingly to the need to fix over too forex frederick women want in so that means

that because the improper integralsfeatures an epic salute to convert as we've shown that in our first example that means that the other from one toinfinity hearing fixed opportu which is that one which is to each of the negativeone-half and that x actually should be a team here canadian county city etc asap dr fixed ii realize that it doesn't matter what the dummy variable is here but imight as well that t go to infinity here

uh... dot is finite in so that means by the computers andtest because the integral over that bigger function because functions bigger because it ishere by faculty there isn't as we have to bea before wanted to g_o_p_ genetics expert over too d_a_x_ convergence eyes we go from one to infinityindividualization is that

he received is in effect squared overtoo and huzi chisimayu takes over too we can see that the inequality reallystarts act extinct balls one and so now if the areaunder this blue car smaller then infinity is in the area underneaththis rate curve must also be smaller than fifty that does make good sense

okay now we're actually interested in four the standard normal distributionactually is the end of the from line is simply toend infinity to infinity so we want to tack on some more area causing i want to go to zero and then all the way over to theleft to the left of my cincinnati attackingon areas not that bad because if you have a function

for which you half the integral from being toinfinity converging well then if you want to tackle and thearea from a to b then all you do is all you realize is that the informationinfinity is you know for a two b classy interval from b to infinity very similar to what we did when we first defined the definitive since the air immunity

that continuous function on a infinite d_n_a_ is smaller thanbeing then the information to the f_ of x_ thea convergence if and only if the interval from b to infinity of apec'sconvergence and moreover when the integrals existswe exists we have the beautiful infinity is equal to the interval from day to be class the interval from b infinity so can be pieced together just like regularintervals

as an example i want to know now if theintegral from make a difference in the futureinfinity to nato explain d_x_ convert resort divergence wrote thisintegral is equal to single from eighteen totwenty two zero two two million fixed-rate over to you colosio from zero to infinity to thinkexpiry of the two by definition we have improper now-familiar infinity toinfinity of columbia weakens ended up in the individualimproper integrals exist india from zero to infinity today to fixwhatever two-d_ exit steam from zero one

e two may have excellent mccue plus youknow from one to infinity determining experiment to d_x_ and dot com virgins integral from make a difference in eighty twozero zero from making trinity dinning if one close enough to make a point to zero and uh... converges basically by symmetry right this is aneven function which means this value here is the same as thisuh...

this value is the same as this valuehere in this values the same this value here so that when converters also and so that means at the end of the from negative infinity to infinity ijaz in a fixedrate of the two d_x_ convergence what other convergent integrals do wehave what we talked about one over except that in fact beach towhat is called the p integral test for integrals over infinite intervals when you've got zero slaughter equalpeaceful article one

then the improper removal from one toinfinity one over x's a p_d_a_ x divergence and forty greater than one the improperintegrals from one to infinity by breaks to thepbx converts so the behavior of the integrals one ofher teaching his exactly at one we had already seen that the integralone over x from one to infinity diverges linden we could say well for

peace more than one taking roots and i will then you have to have to messwith with numbers smaller than what a little bit butbasically wind you have route off a number

that is greater than what have you readit we're going from one to infinity tokyobroke off a number of that is greater than one is smaller than the number which means one over that with thespeakers and a number of some of these integrals rural graders in the in about one of wrecks so they mustdiverge these integrals are slaughter then the integral

off whenever x because when peacegreater than one week take any power power key greater than one off the number and so we would get anylarger denominator and therefore a smaller overall number just because it's so more than oneoverextending its convert in because or speak with another infinity area sitethere but it's still a

pretty good way to just remember things other than simply proving the wholething which we can do because for purely quota one we have the diva from one to two fifty xto the p thats theo extra my history right and so that would be one over negative pto the one takes to that maybe fifty plus one bs the top goes to infinity i guess i could really believe that with index rather thandirty here

and uh... well is now the following wimpy is greaterthan one twenty is greater than when we have a native exploded here and that means that the upper boundthings go two zero two just end up with bond over uh... p minus one custom or bob will be negative one of her native peoples font

whereas twenty is smaller than when weget positive exponent infinite x going to infinity gives us at the upper bound infiniti andso that would mean that the area's and that's exactly what course with therims aspect things cool off infinity p greater than one things converged indus and we have people's one forpeople's one thing that will be seen that that's

the end of the log resumes t goes toinfinity and that's infinity all right uh... so now it's all before thecomparison test let's look at the improper integralsfrom two to infinity whatever x-cart just one p x now that's something yes you know the a directive aboutsaying i'm not going to tell you what it is but that's something that you maywant to try to dig up your mind

humiliated review book this thing youcould already do this directly but let's work it out here at the cove here isn'tit in basically if i'm looking at one oh breaks squaredplus one well dot is only going to get bigger ifi drop the class one right because i have a small originality in so that means and that uh... hardestpart grozny for the computers and test is to figure out what to compare thingstoo but dropping the class one denominatorsuggests

that ought to compare this with the oneover x we're here and when i do that without and just wantto know if you want to break scrupulous once they start pulling one over textsquare and of course defiant go to the reciprocal saw remember whenyou were a reciprocal steve inequality changes i would need to know whether x crackedus once gregory pulls an experiment that is true in so that means to be crossed theintegral from to to infinity whatever

expertise converges so does the you know from tutoring thinganyone ever expect class one individualization again it doesn't say whether the areas areflying at her infinite but what it gives us is that whateverexpertise here and one over uh... expect plus one issue and we cansee that whatever experience greater than one over express class what next example i want to determine if theimproper integral

forex just five over expert months oned_x_ convergence or divergence well let's see where looking and what happens near infinity the five and the one are you going to domuch so this is going to look like for over x will be no sabado breaks dive orders sothey probably want to see whether this thing here is greater or equal overreacts now the things that we can'tdo this we can cross multiplying what we

can do however is ki and realize that four x just fly is greaterequipment for xlr eight square minus one and that's because well we just drop theclass five any if you drop a class five you get something that is smaller and so that means we still have tofigure out where the four xlr expert minus one is greater equipment for overx

forex over expert minus one if i drop aminus one hundred and dominator against mother-in-law later which gives us if i have a minus one of the knowledgerapid smaller denominator which means i get a larger number and when i don't have the minus one and so that means for it so great scriptminus one s greta requirment for its over x weird and that's very cool

and for a breaks because it's indeed the same insult that works and that means because the integral from one to infinity four over xp excitewere just so does the integral office rationalfunction because this fragile function is greater equipment for rex from one to infinity at least individualization is here is four x plusfive or expert plus one here's one of ur ex and we have infinityarea here which means that

the area underneath the great corp mustalso the infinite begin you can see that these find that the united nationsare a little problematic because we don't see whether the recruiters evergoing to come to cross the cooper front this is something that we can only do with the abstract reasoning that we justwent through all right now we could also do merekoestimates if at this stage

you have a little bit of a headache we have looked at a new concept whichmeans that the improper integrals we have to look at compare is intenseweek work with inequalities which is not something so this is all you know this is hardwork she will take a break hit the pause button at the stop buttoncome back to the thing later uh... int at an america west

but what america west of the to-do list if we go back to you comparison test or if we go back to how we're going to estimatethings well if we have something like the function ito made a fixed-rateover to if we just find it cut off so that to the right off the cutoff there's verylittle theory a in the improper integrals basicallygiven by finite here height so if it affects crater goji ethicstriplezero for extra requirment

indiana for a death in the f_ of x_ theextra your calls in the end of infiniti g affects the ex-wife already seen that and so if we can finda cut-off in easy to integrate function has had very little area underneath from eight infinity then our only function will also have been in thearea in this tale

if that means we should be able to do inamerica listed let's see how that goes for the improper able zero to infinity intervening it takes forever to i wantto find a point eight so that the inequality in the gulf made it todetermine if experiment two d_x_ assortment intermediate five holes and that would mean that i get the firstfive digits authors integral as soon as i complete you know from zeroto eight then we want to use

l_a_ computer i resisted compute into before accurate digitally justintegrate from zero to eight well ok so for any query pls i want we have that the into afraid infinity each of thenegative exclude over too is less than a quarter of the incompleteinfinity company x over too we have done that with a little bit ofblood sweat and tears inscri did to compare isn't it

so now we're going to use this stufffrom the computer is attached to actually get notes right so the comparison test is also ultimately numerical tool can be used to so what do we need we need to find age so that the integralto a jury duty teachers in a good x over to the exits or intend to make it fun

if we have that then we know that this integral here definitely it's morethan ten to me alright so we know working out the editor of this and allthat we've done that before we need to find in a civic to into the aid over toit's more than ten tonight at five so the divide by two u_n_ takeover the msfright so we have today if i were to go over them off thathas to be greater than native any over too

and that means that eighty must begreater than they could lead to a time slot rin tin tin tin eighty-five overtoo that's another one of those little trastbecause this looks like a negative number but the more of them off numbers between zero and one is negative so thisis minus minus this is class and this is approximately twenty fourpoint for one to one

and so that's this cut off eight that we want to find now we can use a computer afterassistant compute the logo for twenty four pointfive to fit well we know now therefore that the ill fromtwenty four point five to infinity engineer experiment to d_x_ is one ofthem ten tonight at five if that means if we integrate from zerotwo twenty four point five each of the native expiry over to d_x_

that number is approximately one pointtwo oh five three three one and that's something that use stratton i have a computer outer assist latestsaid give me that in april ended gave me that but we know that the other from church infiniti ofneed to make expiry of the two d_x_ is approximately that number

one point two oh five three three one and that's because irrigate induced tale it's more than ten tonight at five sothey could at worst effect this last digit itself is what we've got to be basically fiveactivities okay now a quick note on function one over who to koi khas anitaexclude over to leaf just computed the data from the irsinfinity to infinity of

from zero to infinity of into thenegative x three over to you if we work this out of the road to prideand this is one of a route to play with negativity to infinity engineer expert over to trusts one of aroutine pride in it was rejected it is a native x screwed over till we knowthis number and by symmetry that's the same as thisnumber so this ends up being kinda all out to spawn a peru to buy aneighbor from new york infinity to

nate's twenty four point five just one of the road to play from eightof twenty four point five two zero whatever route to top i integral from twenty zero two twenty four point fiveplus one of a return flight info twenty four point five and that's approximately the first oneis approximately zero annex was approximately one route twopart-time set one point two five three three one that we got annexed by symmetry ready to natoexcludable too isn't infotrieve

then we have to us all over it to pi onepoint to five five three three one plus another zero because again this wasapproximately zero here it's small bits this intend to make it bicycle pairs of these numbers it's zero and that ends up being approximatelyzero point nine nine nine nine nine separatist approximately one and one of the things institute sticksit in one ups

that these improper including the is equal to one proof ultimately actually requiresdouble integrals and polar coordinates so that's something that we will do when we do multi variable calculus but at leastnumerically here we were able to show that this isapproximately equal okay so now use called absolutely convergent

if it only if the improper integrals of f absolutevalue convergence what c_i_a_ view behind absolute conferobjects new concept right we've got a new heating you no longer in americalistens the idea is that negative signs make estimates many hardit would be much nicer if we could always worked with positive quantities well if we take absolute values we dothat of course

whereby uh... a little bit of trouble here because change the value of the integral is long as we're just interested in whether the integral exists or doesn't exist that in st price that we are glad to peso here's a theater and that's an importantwhich is that if the improper in a pro

convergence absolute peace and alsoconvergence and the proof is whether it's workingwith inequalities basically we will first prove that they will fromaid to infinity of tomb with absolute well we know that the you know from intoinfinity of to with absolute value convert just and that's because absolute convergence means that thistemple without the two exists and thenmultiplying it by two but it doesn't change the convergence behavior

it now because the zero is hysterical fanatical dextrous infasten vedio thanks frank that's either two efa fixed-wing texas positive towardzero when the ethics is negative uh... that's less inventor insistingacquitted two-way topics absolute values that means that the improper integrals for major infinitely if a fixed classffx absolute value converged by comparison test and that means that the you know formajor infinity echo sixty x which is just the over major infiniti ofever pex plastic of excess or value my

seattle elevates absolute value so improperintegrals c alfred works the same way as for regular integrals i couldn't state the xmas sierra minutes it's notthat hard uh... and it's pretty easy to acceptalso now this vehicle is the sum of twoconvergent but the difference of to converge integrals so that one also and that means that

absolute convergence will imply convert however we need to make it look in his firstdetermined absolute convergence uh... determine if the integral co-sign x uh... over executed d_x_ convert asignificant burgess we want to find a value up to an arafat most five timescontinued for croissant makes or breaks q is this a ripple than one over excusebecause the course on absolute values

always mr nichols and one int as long as alexis positive telegrams and one we have that this isreally yes cynical than whatever excuse okay so the integral convergesabsolutely whatever excuse the end of that conversation that meansthe interval office absolute value convergence even so now we do an estimate theintegral of one of ur excused from eight

to infinity they make that smaller thanfive times tend to the negative fine frank if we make the input of the opera balance more thanfive times ten two may two five indianapolis self will also be slowerthan five times the attorney-client indeed distributive whenever x cuba's negative one over two x squared from eighteen finity in that would be one of her to ascrapper infinity begets around in the

minus minus v gives us going over to a squared we need to have to be smaller then five times ten tonight at five oh peaceably bringthat to over giving as a tent in eighty four and then we also got a reciprocal severerealize that is great has to be greater than one over

tend to make it for which just came to the forth and so thatmeans today has to be carried great event swerved into the fourth which is can scarcely must be greater than onehundred so a if we want to make it strictly greatersince we didn't work very clear equals one hundred and one would do itand so the integral from wanted video hussein explore excuse

is approximately the integral from one to one hundred and one quartermakes or breaks q because from one hundred and want to infinity the able of course xlr execute thesmithsonian quote off hussein xlr execute absolute valuewhich is nice and in one of one ever executed which we have seen to be soul there inst subclass zero bright and that'sapproximately if we were set up the

computer out her sister negative zero point zero eight four three seven just something that could only affectthe fifties hit there so this is something fairly accurate at least for the first four digits ofcourse with the five year the fourth digit quote also still be affected last thing i think about absoluteconvergence instead

conversions itself does not applyabsolute convergence that is something uh... that is a little bit technical ithink we're going to do it with pictures basically a look at the function uh... testifying to be one of our k fork slurry clicks market cause one half and make it one of a key for p plus onex moderate quakes smaller take us one what does that look like that gives us erecting the height oneover k on the first half often integer interval int the approach

the corresponding negative rectangle and the second half himself this stress this is zero discuss thisissue of this persistence here this does this is the rest of the current budget cancellation here cancellations here at these things getsmaller and smaller also so that is things go to infinity this thing the dust converge

if you take the absolute value you get one plus one half plus one surplus onefourth plus one-fifth in someone and so on which he is graders then one over exit we know thatthe input of one of rakes diverges which means that the end of the absolute value ofthis function diverged

so we have possible correctly corn virgin improper integral which is so that improper in april does not convert absolutely so just ninety throwing away the negative signs sometimes okay

inside if you want to work that out yeahi i don't think i want to go through the details here let's take a look at them interval of one city f_ of x_ d_x_ absolute values theintegral from one to eight o'clock x from one tothat nearest integer if affects the express the hope from thenearest integer to a itself because of the cancellations that's justthe integral from the nearest integer to eighty

and that's all our equals and one halfone over the nearest into her because it was coolwith one work eight their right that's is eight goes to infinityconverges to zero which means that the improper anywhere from one to infinitylexus function the east the into the from wanted to the f_ of x_absolute value as frederick wheeler-dealer who wantedto make one for x we've seen that in the picture and that is infinite

okay so that these precise argument there the last thing we want to look at the gamma function same remark islamic right made earlieron we're looking at the concert tour we're looking at the marriagegood-looking estimates we're looking at working with absolute values ininequalities and so if you want another break

go ahead and take it right here and thenlook at the cannot function which is interesting because it interpolate suspect or a approach the function cannot offer which heseemed to go from zero to infinity text of the alpha might is funny to me sixtyx which is defined for have a greater thanzero is called gemmell function why is that the important but we'regoing to see that in the remainder of this presentation

this function is defined for alpharhetoric will then one because what we have here then would be aches too some power times he does in a fixedindicated he showed the x_-two some power times each is a native x dot isn't a conversion integral we will also see once we have improperintegrals over singularities because of course for dofus more than one week thatexterminated something with which blows

up at the northbound we will also see that it's defined for zero smaller office more than one but what his enemies v interesting is for all alpha greaterthan zero as long as the computations were going to do we have to gamblesalpha plus one ezell clinton's gamma class gamble fail thought

and those of course something thatthat's like what the factorial function as like the fact order function always multiplies the stuff that happened already for every positive integer in we have a camel thin is in minus onefactorial right uh... in we will prove that so proof for alpha greater than zero the cannotalpha plus want to see a concerted

aches to the alpha plus one minus one into the night affects which is of course you know from zero toinfinity x to the outfit into the need to fix invest something that we can do you with integration by parts we keep the first n integrate the secondso we can't make it to fix to the alpha determining effects from zero to infinity minus the into the from zero toinfinity

made a few dozen eighty six times thederivative of this thing which is l flakes to that alpha minus one if we work that out then at the front we get zero well yet we're going to getthrough because esti goes to infinity the exponential goes to infinity int at zero well the clinton as long as alice kramden zero we getzero to a positive powers of its zero minus minus gives us last

alpha comes out in this issue noconcerted finney what's next aches to the alpha minus one heat of thenegative x and that's of course it cannot function off alpha so this is alpha times gandalf health so that means no gavel not so now let's take a look at it so that proves this equation right he's now let's takethis up with the factorial function

gemmell one is the integral from georgiathrough the x_-two the one minus one easily fix which is you know fromgeorgia canadian companies x d x which has made a speech is a native fix from zero to infinity and that's what because essex goes toinfinity beget zero net that we're going to get minus minus wilkie so gamma often

is kendall in minus one plus one which is in monday's one by what we havehere in last one p_m_ often minus one which is in my eyes one-tenth in lettersto get off in my list to let you just keep applying he questioned your injured keep doing that until the end up within months oneinman's too is a little bit down to two attachments camel one

and you know that gandalf one is one so this park here we might as well dropsince this is a product that we have left use one turns to times and so onall the way to in minus one in fact is in minus one factorial in fact finishes the result and so basically thevisualization iis throught the factorial function

then or because in minus one factorial so we now we remember zero factorial es one one factorial es one tool factorial es tu three factorial essix remember it's all shifted by one the cannot function just intro provides these points and thoughthis why he cannot function is also called

continuous factorial function so this is a function that interpolate stacked orioles it is quite important in statistics and in various situations in the few need to interpolate for factorialfunction as because it fits very nicely with improper integrals because it is a

grant application of improper involveswe present at that right here also you've heard a lot of stuff aboutimproper integrals over infinite intervals you've got it good beautiful but you did to makeeverything settle in some of the work that goes straight with the definitionof likud reinforced integration skills convergence tests as well as in americaestimates as well as absolute uh... trained you were in working withinequalities which i keep saying it whenever it comes outthat it's something that is on the whole

prices estimates are very important in the actual application of capitasimply just have to train ourselves in the requisitepattern recognition is you do that i'm going to take a break i think ifrequented something like op six presentations over a twenty four-hour period were somethingmore and a twenty four-hour period

so i'm going to give myself a break where as you well doing the homework you just break ifnecessary intent continue with the pays off the class i'll see you in the next presentation