 Welcome back we are at lecture 44 this in-class group of brilliant students took their test on Friday today is Monday so we are I hopefully we're not in morning over that I've heard a few comments about it but I don't think that it knocked anybody's feet out from under them that was not the intention I'll have a little bit of grading time today but most of my grading time probably this week will be on Tuesday so I'll try to get these back to you by Wednesday if I get some graded I'll return what I have graded tomorrow but Wednesday would be more realistic we will continue chapter 8 with more convergence tests you have that summary sheet so we have moved from the front of the summary sheet to the back of the summary sheet we've just gotten started with the limit comparison test by the way there was one of those on the test you could try the comparison test but it the denominator was larger and it needed to be smaller in order to compare it to an existing divergent series but it was kind of it was close close enough to then bounce to the limit comparison test can't remember exactly the problem one over three in plus two I think so the plus two meant that the denominator was larger therefore the value of the term was smaller and it needed to be larger in order to diverge all right so we will encounter not necessarily for the first time but at least as far as tests are concerned convergence tests for alternating series in 8.4 we'll look at another version of the harmonic called an alternating harmonic different kind of decision as far as convergence what else do we have we have absolute convergence if a series converges absolutely that means that not only does the alternating version converge but the non alternating version also converges so in other words it doesn't matter if it's alternating or not it's absolutely convergent and some of them are that way others of them the alternating version converges the positively termed series that is paired with that diverges so it is not absolutely convergent and I don't know if we'll get to ratio test today that's realistically probably what we'll do in here tomorrow is take our first look at the ratio test and under what conditions we get a conclusion and when does the test itself fail so we are in chapter 8 section 4 why don't we just call it other convergence tests and we'll just kind of continue to add convergence tests and also before we finish we'll get some power series and then certain types of series called McLauren series and Taylor series determine when they converge and then applications of those and then we'll take a summer vacation how's that sound because that'll be the end of chapter 8 okay alternating series so what do they look like you will see a term that looks like this in there it'll be to the end power possibly and then we'll have something else in here let me just say it's right now some description of how to get to the nth term if we were expanding this one would the first term be positive or negative negative because negative one to the end I don't think I want the first term to be negative I want the first term to be positive how could I correct the description of how to arrive at the nth term we started at zero or we could just change that up one or down one to get the value we want to make this start at a positive so do pay attention to the exponent because it'll tell you if it starts out with a positive term or a negative term and we know from there they alternate so the first term here would be positive a sub one whatever a sub n is it'll vary from problem to problem and so on so we have this alternating series we should not expect alternating series to behave the same way that the series would behave if the signs were not alternating obviously it's going to make a big difference into how these terms add together this is one of the easier ones the alternating series test in order for an infinite series of alternating terms to converge two things have to be true the n plus first term and let's just say this is for that's the way it's written out in the book so the n plus first term forget the alternating sign part of it that's taken care of here just the part other than the alternating sign is smaller than its predecessor so in other words it is decreasing and it's technically not just decreasing it'll be ultimately decreasing so if the first couple terms kind of bounce around one's larger one smaller again disregarding the positive and negative signs we're talking about the magnitude of the term it doesn't really matter but ultimately the thing must be decreasing as we go and if you go way out to the right the limit of the nth term as n approaches infinity is zero that's it if those two conditions are met then the series is converted so it's got to be ultimately decreasing in magnitude the magnitude of each term disregarding the plus or minus sign and then eventually the nth term has to get closer and closer to zero so let's take an alternating version of an old friend harmonic series we know the harmonic series when it does not alternate in sign diverges we've seen that a couple different ways probably well three ways we did a grouping of terms and we could group the first one the second one the next to the next four the next day the next 16 each time it was greater than or equal to half we integrated it it diverged and we kind of categorized this as a p-series and one over into the p and p was one so we had three good reasons why the harmonic series diverges now we have an alternating harmonic what's it look like what's first term one it's a Monday okay get it get it moving a little bit I don't know about you but I'm kind of glad to be back at school and work because all I did was dig holes and plant bushes and plant trees this weekend so this is this is wonderful I like being here this is a safe and much more kind environment what's the next term negative half now we've got it going what's the next one positive one third minus one fourth and so on so it is harmonic but it's alternating in sign what do you think converges or diverges it's going to converge so our first part of the alternating series test is the I'll go ahead and write it out you don't have to write this out so if we look at the n plus first term disregarding the alternating sign well the n plus first term would be one over n plus one is that less than or equal to one over it is right larger denominator smaller in value so that's true statement is that true does the value of the nth term as we move way out to the right as n approaches infinity is it disappearing to zero that's true so the alternating harmonic series is convergent this this might be the wake-up we need here this morning any guesses the value to which it converges it does converge we just verified that with this two-part test that's a little tricky so let's look at some sequence of partial sums the first one I can do that even on a Monday some of the first one terms that would be one about the sum of the first two terms one half some of the first three terms so we're just adding on this one right so it'd be a half plus a third which is five over six some of the first four terms so now we're going to take this one and get rid of one fourth right so what's five over six minus one fourth how many seven twelve let's see if we see anything happening so some of the first one term is one probably too much right that's probably more than the actual sums going to be so then we subtract a half and the sum of the first two terms is a half that's probably not enough meaning that we probably did what we probably subtracted too much so this was too much this was not enough so when we get something that's not enough what do we do we add something back in guess what we add too much back in so this is probably too much so because now it's too large what do we do when we incorporate the next term we subtract some away guess what we subtract too much away that's the way the whole sequence of partial sums goes this is too much this is not enough we add in the next term it's too much we subtract some away it's not enough so we continually kind of trap where the actual sum yes it's somewhere between one and a half now it's somewhere between a half and five six somewhere between five six and seven twelve and you continue that process that's pretty much how all alternating series work let's take a look at another one to try to do a convergence or divergence test and then we'll look at kind of how we approximate sums of infinite geometric series so does it converge or does it diverge so again ignoring the alternating signs what would be the n plus first term to the n plus one five to the n plus one plus three that's the n plus first term in magnitude the nth term that's true and if so how would you kind of talk through that one how would you justify okay so the both got larger didn't this numerator get larger by a factor of two the denominator got larger in essence by what a factor of five right so is that enough to convince you that it's smaller than two to the n over five to the n plus three the plus three is kind of irrelevant in this problem kind of tells us where we start with the position of the denominator but as far as determining larger and smaller it doesn't have anything to do with that so multiplication by two up here and multiplication by five down here the denominator got larger faster than the numerator that makes it a smaller fraction so that is true might have a problem here does the value of the nth term disappear to zero as in gets larger and larger and larger no so eventually the value of the positive terms are going to get closer and closer to what what is the answer to this two fifths they're going to get closer to positive two fifths the negative terms are going to get closer and closer and closer to negative two fifths but we need for them to disappear in order for this particular series to converge they don't therefore this is divergent this truthfully I don't know about I wanted to make sure we looked at an alternating harmonic I think that's in the book I'm not a hundred percent sure of that this is a problem that I know for a fact is in the book the first part of this is a little tricky to deal with so I want us to look at a procedure of how we can determine if the series is ultimately decreasing which is part one of this two-part test so again we're ignoring I mean we acknowledge that it is alternating it is and then we kind of set that alternating piece aside in order to do the test that might be a little tricky to deal with if ever you're trying to determine if the function associated with the nth term description is in fact decreasing I mean this is a calculus class how could we use calculus to determine if in fact the thing is decreasing take the derivative and it should be beyond some number two five eleven seventeen beyond that point it should be negatively signed right so the function associated with this and I'll put some question marks because that may not be quite as clean as the one we looked at before so this is always a weapon take the derivative see if in fact the derivative is negative for some point in time forward so denominator times derivative of numerator minus numerator times derivative of denominator all over denominator squared what do we get out of that 2x to the fourth plus 2x minus what 3x to the fourth that look right so we've got some x to the fourth terms two of them minus three of them so we've got minus one and in the numerator we could factor out an x so how do we go from here to establish when this first derivative would be negative because we want to make sure it's decreasing are there any obvious critical values of the first derivative x equals zero we don't really care what happens on the other side of zero right because these are in is the number of terms so x is the number of terms so x equals zero is a critical number where do we get another critical number Q Brut of two we would set each factor in the numerator equal to zero we could set each set the denominator equal to zero because the derivative could be either zero or undefined I think we're going to get a negative value down there so that becomes irrelevant so what do we have here Q Brut of two so we could do a number line test is it the denominator always positive right so we don't really need to plug values in there and we don't need any values over here to the left of zero because x is the number of terms so let's see what happens between zero and the Q Brut of two what's the Q Brut of two so we could put in one right if we put in one that's positive that's positive denominator is positive so the derivative is positive so in that vicinity of this curve of course we don't have a curve we don't have all those points we don't have a continuous set we just have dots but in that area the curve is increasing how about larger than the square root of two or we could put in two that's positive that's negative right we put in two for x that term is negative so every other term is positive that term is negative which means from that point right because there are no more critical values to the right of two this thing is decreasing so is it an ultimately decreasing curve this verifies that it's ultimately decreasing so from the square root of two which for us is two and larger so from the second term onward to the right this thing is decreasing so this function is decreasing therefore the series not the alternating part but the just the size of the term so the series in magnitude term by term is also decreasing so back to this is this a true statement apparently it is okay although it may not always be an easy one to try to reason through in this form if that is not easy go to the first derivative verify using the first derivative that the curve is in fact decreasing alright that's part one it is ultimately decreasing and the second part is the limit is that equal to zero that's true okay not anything real complicated there but the first piece you can always resort to to calculus to validate that it is in fact ultimately decreasing if you have to kind of say by what test well by the alternate alternating series test that's what this is and it only works for alternating series alright the rest of this for alternating series deals with how is it that we approximate the sum and it will be approximate it's not like infinite geometric series where we find the first term and we find the ratio first term over one minus ratio that is exactly the sum of those terms all the way out to infinity so we're going to approximate let me read the wording that's in the book actually let me show you this first and then I'll read the wording in the book did it put this up here before but I think we've we've done this so here is the alternating series test notice their description of the nth term negative one to the n minus one that's okay that this starts the series out with a positive term so they alternate the value of the term the magnitude of the term is B sub n which is always positive regardless of the final outcome of that alternating series so there's our first test is it ultimately I guess we could set not necessarily for all in right but for in beyond a certain point so that all the rest of them are in fact decreasing so we've done that so how are we going to approximate the sum so the alternating series estimation theorem so it's got to converge so it's got to pass this test in this test so this is the remainder which we decided earlier in the text was really an error associated with the sum of adding the first n terms together notice it's absolute value so we don't know if we have too much or too little but we're off by that amount so that's the difference in absolute value between the actual infinite some and the sum of the first n terms you're not going to get one that's easier than this as far as determining error what is that so what is going to be the difference between the actual some all the way to infinity and our sum that we added together up to the nth term what is that isn't that the n plus first term right so I don't know if this says this to you but it certainly says this to me the error or remainder is actually less than or equal to so here's an upper bound for the error all we have to do is pick off the next term we added the first 11 terms together how far off might we be the value of the 12th term we added the first 22 terms together how far off might we be from the actual sum the value of the next term which is the 23rd term so all you have to do is basically look at the next term and you know how far off you might be in the in the maximum sense in the worst sense we don't know if we've got too much or too little that's not our issue to decide the issue is how far off might we be the proof of that is basically rooted in the fact that when you take the positive term let's say is the first term and you throw in the negative term in all alternating series by subtracting something from the previous partial sum you overcompensate for where the actual sum actually is so we had one we subtracted a half we ended up with a half the actual sum was somewhere in between we under the actual sum what do we do we add something back in one-third low and behold we add too much back in every alternating series progresses in that fashion there is a nice diagram on page 587 that more or less explains that without numbers but we've already seen that phenomenon in numbers they've got letters on that page but it's the same thing in diagrammatic form that you gradually converge between your two most recent values you gradually converge on the actual sum with these alternating series the wording in the book is really kind of cruel so I don't use that wording the next theorem says that for series that satisfy the conditions of the alternating series test so we've got to make sure it converges the size of the error is smaller than b sub n plus one we have that so I think it's pretty kind in wording to this point which is the absolute value of the first neglected term I just don't like to use that it just sounds kind of cool that you're neglecting that term so I'd like to keep it a little more positive that it's the next term it wasn't neglected it just happened to be the next one in the series so I don't like to use neglected because it just I'm afraid it might hurt itself esteem and it would just be sad so I won't use neglected term from this point further it's just the next term so we want to sorry probably is not good for a Monday maybe that's good for later in the week but basically the next term we have the sum of the first six terms we add them together we take a calculator we get a common denominator however we add them together we know this what's the error associated with the sum of the first six it would be b sub seven which is the absolute value of the seventh term the next term actually it's the upper bound for the error the error is actually smaller than that but that would be the error at its worst so to speak so let's take our alternating harmonic we know it converges we've already verified that so let's say that we want I'm just going to kind of make something up let's say we want the error to be let's not approach it that way let me approach the second example that way let's say the sum of the first I did this six terms up here so we're going to stop it it's not an infinite series it's truncated at n equals six now we added this up to what some of the first four a couple minutes ago what did we get 712 so now we want the sum of the first five so we would do what take of the sum of the first four and add in the next one which would be one-fifth what's 712 plus one-fifth 1960 1960s I think that sound right 19 when we have to multiply here by five and here by five here by 12 and here by 12 so there's 35 47 60th and if that's I'm sorry that's not the sum of that's the sum of the first five we want now the sum of the first six which would be 47 60th plus the next term what's the next term minus one-sixth 37 as a decimal what is 37 60th and we don't know what level of accuracy that is at this moment but now let's address that this is not the exact sum what is the so-called maximum error for this sum and by the way do you think it's an under estimate or an over estimate under under because we just subtracted right we subtracted too much is the problem so we're under the actual sum so what about this r sub n in this problem so the difference between the actual sum and what we now have is the sum of the first six well what's the size of the next term what would it be we just added added in a negative one six the next term is one seventh what is one seventh one four two nine so that's our error at the worst actually our error is smaller than that so we're not actually even though we added together six terms this is not a great approximation why is it not a great approximation because this series kind of goes kind of slowly from one sixth which is what point one six right repeating to one seventh which is point one four two nine so it because it plods along so slowly we'd have to add together a lot of terms to get a pretty good level of accuracy but at least we know where we are we know we're off by at most that amount let's take a little different approach to this problem where we are kind of handed a level of accuracy I'll probably need some calculator help on this one so we want to sum that is accurate to three decimal places what's this look like what's first term one next term it's negative right they do alternate one eighth see the difference that the other series they weren't changing that much as you work your way to the right we're getting a significant change as we're working our way to the right so if we wanted a sum let's say we wanted the sum of the first four terms wouldn't we be off by at the most one over 125 according to that alternating series estimation theorem that's not three decimal places of accuracy yet is it what's the next one six cube 36 times six 216 is that sound right think we're there yet three decimal places of accuracy what is one over 216 probably not there yet right because we've still got something occupying that third decimal place we want it to not affect the third decimal place next term whenever 343 what's one over 343 point zero zero two nine probably still not there yet we need to keep going so that's one over six cubed one over seven cubed one over eight cubed what's 64 times eight okay we're getting pretty close right so so far we've got the sum of the first eight and what's the next one and what is one over 729 so we want what nothing there you have to have nothing there that's going to affect this accurate to three decimal places that's a that's a good question if you have one there we had one there we'd be all right yeah what about other another number occupying this position we went from two to zero being in the next place that would be okay too so we want accuracy to three decimal places so we could have one there and be okay excuse me where are we right now isn't that one excuse me isn't that one gonna affect the third decimal place though it is right I really think we need to keep going till we don't have an occupant of that place what's the next one sorry one over a thousand we're about to end this class a little early sorry so we have how much I think we're good there right so where are we here zero zero zero what seven so if we could add the first ten terms together I would recommend a calculator rather than get a common denominator like we did before the sum of the first ten terms should be within this amount of the actual sum all the way out to infinity we're actually a good stopping point since I can't stop coughing we will resume at this point tomorrow