um hi class uh uh today or uh this section uh is uh about sequences so we're going to be covering 11.1 sequences and there are some videos here that i'd like you to watch and this is also part of your assignment on canvas but i'm going to go ahead and explain the concepts that are behind sequences so we're going to start first with the question of what is a sequence so a sequence is a discrete function so we have the sequences the discrete function discrete means uh points where the domain is a set of positive integers positive integers will be 1 2 3 4 5 etc and and the domain is given by a formula or a rule for example the function f of n where n is a natural positive number will be equals to 1 plus n over n squared um usually we call sequences by the letter a and the subscript n and we say the sequence a sub n equals 1 plus n over n squared so when n equals 1 that's our first integer positive we plug in the 1 into the formula or the rule which is given by a sub n 1 plus 1 divided by 2 to the 1 power because n equals 1 that will be 2 to the 1 power correct actually i have to fix this a little bit so um i had to fix my first example the denominator should be 2 to the n power and so the a term corresponding to the value n will be 2 to the n so when our domain n equals 1 a sub n or the value of the first term of the sequence would be one plus one because it's one plus n which is one divided by two to the one power so the answer will be two over two which is a one and then we proceed with n equals 2 2 will be our domain our next positive integer 1 plus 2 which is the value of n at this point divided by 2 to the n that would be 2 squared that'll be 3 over 2 squared and when n equals 3 as our third domain element and we will have a sub n which is our uh third term in the range will be one plus three because n according to the formula equals three and two to the n which is two to the cube etc etc so we create a sequence or a list okay and uh we keep on going the question that i want you to have in mind now is is the sequence is this list that we have decreasing or increasing so the list that we just obtained would be one divided by 3 over 4 divided by 4 over 8 etc etc would that be decreasing or increasing basically is our sequence this list of elements decreasing or increasing also is this list bounded bounded means uh is in between two values of two numbers that contain the entire list or the entire sequence or unbounded means that this list goes to infinity it cannot be contained in between two numbers all right those are the questions i want to have for now in mind let's go over question number two um so again this is a worksheet and then i'm providing uh some of the answers and uh on this worksheet question number two i'm going to go ahead and explain the answer but these answers are in your um in the powerpoint presentation for this section also in the book but i'm gonna try to help you with some of the answers for uh the worksheet uh our question number two says expand four terms of each sequence below the first sequence is a sub n over n plus one in this brackets and n goes from one to infinity another way to rewrite the sequence is to say a sub n equals n over n plus one no needs to indicate the bounds or the values of n that we're going to start with so just like before we're going to create a sequence list in a list of numbers our domain is n equals 1 n equals 2 n equals 3. as we start with n equals 1 we obtain a sub 1 which is n equals 1 is 1 according to the formula or the rule and then 1 for n is 1 plus 1 is 2. so the first list the first term sorry of the list will be one half the second term will be when we work with n equals two so a sub two will be n equals two divided by two which is n plus one which is two over two plus one which is 3. so the next element in our sequence will be 2 over 3. when n equals 3 right our third element in the domain our element in the range will be given by the rule by the sequence rule which is n which is three three plus one in the denominator which is four so that one will be three over four and the list keeps on going to infinity and this will be the range and we're going to graph this this graph the sequence so we can observe how this is a function so we have on the x-axis that would be the n-axis this is where we have the positive integers starting at one we always start a sequence at one unless we are told otherwise and on the y-axis is the terms a sub n so when n equals 1 that would be a sub 1 which is a one half and then as n sub n equals 2 we have 2 over 3 which is about 0.6 when n equals 3 that would be a sub 3 which is 3 over 4 which is 0.75 so as you can see this values of a sub n as n goes to infinity is increasing increasing increasing but i'm not going to infinity these values we see here are approaching they seem to be approaching the number one and so we're gonna we're gonna try to understand that behavior and so one way to do it is we're going to take the limit of the sequence and we take the limit by setting up a limit and we say limit of the sequence n over n plus 1 as n goes to infinity and we just divide by n each one of the terms and we get that the limit is the same as the limit of one over one plus one over n and then goes to infinity one over one plus zero this is one the limit is one so there's uh this list the sequence which is the discrete function approaches one it will never get to one it would approach one so that's the limit of the sequence so when we say limit of the sequence is when n goes to infinity there is a second way to find this limit using calculus we apply the limit optimal rule because n over m plus one as saying goes to infinity uh s of the form infinity over infinity if we have infinity over infinity infinity over infinity we could apply l'hopital's l'hopital's says that this limit equals the limit as n goes to infinity of the derivative of the numerator divided by the derivative of the denominator derivative of n with respect to n is one derivative of n plus one with respect to n is one this is the limit of one over one which is one the limit of one is just one that's the limit again okay now number two this sequence is bounded okay bounded means is in between two numbers so it's bounded below and above let's check the bounds all these points in this sequence are less than one because of the limit let's just found the limit so any a sub n is less or equal to one actually it's gonna be less okay so let's change that it's gonna be less than one but so we have this is less than one let's fix it we're fixing that and uh but uh when we uh when we uh evaluated the value of the first term which is n over n plus one we used n equals one so when n equals was 1 that would be 1 over 1 plus 1 that would be 2. so the lowest value of a sub n is one half because the other elements will increase so um so we have that um this limit right this limit is going to be sorry the values of a sub n are between one half and one so that means that we have a lower bound and an upper bound okay we have an upper bound and the lower bound let me just fix this inequality there you go okay what else do we have um so let's just do a recap uh the limit is one the sequence is bounded below by one half which is this first value of the sequence when n equals one is bounded above by the limit which is one also we need to understand that this discrete function which is made up of this dots right this points is part of a larger function this larger function is the associated function to a sub n that function is in this case x over x plus one all right because when we do derivatives we're doing derivatives of on n but we're really doing derivatives of on x because it has to be a continuous function the one that we do derivatives on so the function will be the one in blue and the sequence would be the dots or the points and also we notice that this function is increasing and also our um sequence is increasing so whatever the function does so does the the sickness okay so the sequence follows the function let's move on to b so in b again we have to um expand the sequence and we have a new sequence now um the sequence b is uh peculiar because it has a negative one to the n power so times n plus one or divided by three to the n notice there is no bounds that means that n is going to go from 1 to infinity and or we can change that we can rewrite this as without the brackets as a sub n equals negative 1 to the n times n plus 1 parenthesis divided by 3 to the n so we go ahead and do we start with n equals 1 and we substitute the n equals 1 into the formula for the sequence and we get negative 2 over 3 n equals 2 we will get a positive here a positive and a positive value of 3 over 3 square negative 4 so we're alternating the signs you see the negative 2 positive 3 negative 4 so i'm plugging in the 3 into the formula given by the formula of the sequence as a negative n plus 1 would be 3 plus 1 would be 4 divided by 3 cubed and then 5 because it's 4 plus 1 because n equals 4 to the 3 over to the fourth power here actually that will be to the fourth power so i need to fix that 2 to the fourth power okay what do you notice you should notice that um the values are getting smaller and smaller even though we have an alternating series even a sequence sorry even though we have an alternating sequence these values starts at negative two over three the next time will be a positive three over nine which is the positive very smaller in absolute value than the previous number and then the next number we get is a negative what is an absolute value is smaller than the previous value when n was 2 etc etc so every element which comes up next it wouldn't matter if it's positive or negative if you're getting closer and closer to zero so when we have an alternate in sequence okay and we want to see if the sequence will converge in this case we took the absolute value of the terms and uh and we took the limit as n goes to infinity taking taking the absolute value of the sequence means that the negative one to the n in absolute values will be just a negative one absolute value this will be a positive term so we're working just with the m plus 1 divided by 3 to the n and so that's the limit that i have in front of me now now that i have removed the negative part or the part that makes this positive and negative and i'm going to use the l'hopital rule one more time derivative of n because this is i'm going to use l'hopital's rule before i continue because this is infinity over infinity in form derivative of n plus 1 is one derivative of three to the n is ln of three times three to the n so that will be a zero as n goes to infinity the denominator will be going to infinity and one over zero one over a larger number is zero so that's our limit okay um i'm gonna leave the next one for you to finish uh that will be c this will be question question c and part two okay um moving on we have question number three um we have a recursive sequence so what is the recursive sequence re re means to redo three is again okay like rewrite okay um represent re means present something again cursive cursive is to write so we're gonna use sequences that require the rewriting some rewriting all right so once again we're going to be working with sequences which require a re-writing of a previous term of the sequence so let me explain that with an example the fibonacci sequence okay uh it starts with fibonacci one he will not shoot two there's two entries these two first numbers are one one so in the sequence we start with one comma 1. the formula for the f sub 3 the fibonacci number number 3 is f of n minus 1 plus f of a minus 2. so they're asking us to add the last two terms the previous term and then the term before the previous since where and n equals three we need f of n minus one which is three minus one is two and n which is three minus two that would be one so we need to one actually one if you're gonna actually two we wanna ask you one fibonacci two would be one plus one which is two so wait a minute so to to obtain the formula for the next fibonacci we have to do a recursive now we have to rewrite we have to write again the previous term the previous two terms so this makes this it makes this uh sequence a uh a recursive sequence okay so one plus one plus one is two i go to i wanna find f fibonacci number four that number would be three because it's f of n minus one so four minus one is three so the fibonacci number number three fibonacci number number three was the two that i had just got in before so i need to re-curse rewrite the previous term of previous terms and i need f2 f2 is a one so i have to re recurs re rewrite egg one so two plus one is three next one would be five and this f5 means f4 f4 is three okay and i'm gonna reuse and i have to use again the number three and the previous fibonacci which is the two so the two comes up again so we have a five and then when we get we want to get f6 we need f5 plus f4 now we five plus the previous one number which is three so previous and before previous that would be five plus three equals eight so this is a list of numbers that we're producing one one one plus one two one plus two or two plus one three three plus two five five plus three eight a plus five thirteen so this is a recursive sequence okay a list of sequence limit uh is going to infinity so the limit of the sequence is uh infinity so is is not bounded okay the limit as saying goes to infinity of the fibonacci numbers is infinity not bounded okay because the fibonacci numbers will uh diverge right average means we're gonna go to infinity okay they might have a lower bound which is the lower bound is one but that's not enough the lower bound here is one but we need actually two bounds we need a lower bound and an upper bound so is not bounded let's continue um number four i want to leave that to you to to do it we actually did it already in the previous example so number four you are uh you will do it on your own okay number five also was done on the previous example that i explained so please go over number five and explain this on your uh mathematically basically using the limit number six wanna do number six so what is a convergence sequence convergence sequence is a sequence which whose limit would be a number okay and divergent sequence is a sequence whose limit will be infinity or negative infinity some form of infinity so examples will be 1 plus n over 2 to the n this is our first example that we worked on so we do the limit of 1 plus n over 2 to the n as n goes to infinity this is of the form infinity of infinity for infinity over infinity so i could do capitals the answer will be the derivative of the numerator which is a 1 plus n is just 1 derivative of the 2 to the n with respect to n is ln of 2 times 2 to the n so it's 1 over a large number is 0. so this is 0. so in this case uh this is a sequence which is convergent to zero okay and mostly is bounded okay it's gonna be bounded because i plug in n equals one that would be three three over two so three over two is the upper bound and the lower band will be zero because zero is smaller than three over two let's move on to a sequence which is divergent so two to the n divided by 1 over n so i do the limit as n goes to infinity this is in the form infinite over infinity again this infinite over infinity so i'm going to do l'hopital's l'hopital to solve the limit so the derivative of 2 to the n is going to be ln of 2 times 2 to the n derivative of 1 plus n with respect to n is 1. this is going to infinity divided by 1. we have infinity so there is no limit all right there is no limit or the limit is infinity or the the sequence is diverging okay um this is just a note um again just as a reminder when i work with this function uh which is discrete i'm really working with the associated function which is 1 plus x over 2 to the x i'm really taking the limit as x goes to infinity because when i'm doing l'hopital's i'm using the fact that the function is continuous and it is differentiable and that is not possible in the function 1 plus n over 2 to the n because this is a discrete function so when i do the work with ends really i'm doing so i don't so i save time i don't have to rewrite this uh sequence as a function where i find the limit of the function that tells me the limit of the sequence i just go ahead and i use a straight the n values okay but technically we have to rewrite the same uh the sequence uh as being inside this associated function when we take the limit as x goes to infinity of this function we get zero and then we can say what the limit also for the uh sequence will be zero okay so we're just saving time by not having to go to the x values number seven we have a precise definition of limit so what i'm going to do is try to explain what this is in math i want to change it around i'm going to have a graph that explains so a sequence a sub n has a limit l and we write the limit of a sub n as n goes to infinity equals l okay or a sub n goes to l or wants to become air which is a limit as n goes to infinity we just saw some examples of that okay this is more abstract what does this mean it means that we're gonna play a game okay this is kind of like a game with a number epsilon epsilon is just an extra number that somebody decided to use for the definition so we're playing with epsilon epsilon is a number which is always greater than zero okay now i want to see where this epsilon comes from and how we can observe it in action in this limit definition but we need to know this definition of limit in a precise way so for every value epsilon greater than zero we're going to be able to find a large n and uppercase n so this goal these two not these two letters so these two numbers go hand in hand the epsilon we're going to be giving an epsilon a number let's say let's say 0.9 then because of that 0.9 it given to us and because of the formula a sub n we're going to be able to find an integer in and it lives on the x-axis okay n lives on the x-axis what is the relationship between epsilon and n when i find a value on the x-axis where when we plug in those values into the formula of a sub n whatever comes out of there okay any a sub n that comes out of that after the n has been fixed okay so let's say for example the big n that we find is a thousand so epsilon is 0.9 and the n is a thousand that means that when we plug in the value of 1000 in the in the in the formula of the um sequence the difference between these two the distance between this a sub n which would be a sub a thousand a one thousand minus a limit has to be less than epsilon so maybe it doesn't make sense as well i'll have a picture here so let's go over this definition using the diagram so these dots in pink are my sequence all right you notice that the sequence is going up and then down and at some point it starts converging at some point it starts converging to the value of l so this is the limit okay so at one point the the function of the sequence was just up and down and then after some point and after some number in you see that i'm gonna look at i'm gonna look at that on the on the on the x axis so the x axis has the end value so at this location after all these n values that i have on the x-axis the elements of the sequence are trapped they're trapped in this corridor you see this corridor this corridor is like like an alley right all of them are trapped okay so the question is how wide is this corridor okay for this value of n whatever it could be it may be a thousand or two thousand um what is that epsilon that produce that that band right that is stripe without a strip in here that would get all the rest of the [Music] elements of the [Music] sequence right and in this case oops in this case you can see the epsilon going up a little bit and going down a little bit okay so from the air we go up and then down so i have an example here so in this case all right i have the epsilon to be 0.9 i just made it up it's also 0.9 so i have my sequence and when n equals 10 i'm able to catch every single element of the sequence inside this strip okay so when when epsilon is 0.9 somehow after n equals 10 n equals 11 n equals 12 n equals three all those elements of the sequence will be trapped inside this there's uh this corridor okay now can i do that again with a different epsilon yes when epsilon is 0.2 all i'm doing is making this corridor tighter tighter so i'm going to be able to catch all the elements right all the elements of the uh sequence to be inside the corridor but not at the same end but at a different end like n equals 20 for example so when i so when i use the epsilon 0.2 i'm going to get the entropy 20 and i can do this all the time so if i get even a smaller epsilon 0.001 i'd be able to catch all these elements of the sequence later and then if i have a different epsilon i will always get a different n if i can if i am able to do this all the time for any epsilon i will get an n then that means that the function of the sequence sorry is has a limit at l okay moving on and number eight we have this limit properties of the sequences and we have the limit of the sum of two sequences is the sum of the limits the limit of the difference is the difference of the limits of the sequences uh this is similar to the limit of a function because these are discrete functions so the same limit laws apply okay we have that the limit of a constant times sequence equals the constant times the limit of the sequence the limit of a product of two sequences equals the product of the limit and the limit of the quotient equals the equation of the limit and finally the limit of a sequence to the power p equals the power p of the limit a sub n and then p has to be positive and a sub n is also positive the requirement for this is that the limit of a and the limit of b would exist or be convergent or okay c is a constant so this limit if we want to work on this when i use this formula for this limit law this limit has to be a number and the limit of the second the sequence here also this one sequence has to be a number so let's go over an example you're in charge of the ones that we don't get to do so the first one is for the you can um go to the book and you can go to the powerpoint okay so the limit of the sum of these two sequences will be the sequence sequence sine of n over n plus five n squared plus five sorry five n squared plus n divided by n squared plus one so this will be the limit of the sum that would be the sum of the limits so sine of n divided by n when we take the limit as n goes to infinity will be one the negative five n squared plus n divided by n when we take the limit that would be five so these two limits exist that means that each one of these limits is convergent that means that i can use this property or this limit property of the sum of the two limits which is six okay it's a side note in here where i show how the limit of sine of n over n as n goes to infinity is one basically is by the l'hopital's one and i just slope it over there okay and the limit of five n squared plus n divided by n squared plus one i'm just going to divide by the highest power of n's and that will give me five plus one over n divided by one plus one over n and that will be a five over one okay so the questions that we skip you are still in charge of them someone escape all the way to number six okay this is an example for number six so i have the limit of n over n plus one quantity to the fourth power so according to the limit property and i can plug in the limit inside this exponent and we have the limit of n over n plus one and then goes to infinity to the fourth power the limit is going to be one of n over m plus one we have seen that in the previous example so the answer is one note the limit of sine of x to the fourth power as n goes to infinity is not it is not the limit of sine of x as n goes to infinity inside the fourth power like before is not y because the limit of the sine of x as n goes to infinity does not exist how come it doesn't exist because when you have this sign is the sign of n the sign is gonna be changing values from negative one to one but it will never equal to a number it will never approach a limit or a number so this limit does not exist we cannot use the property of limits for sequences okay let's move on to number nine the squeeze theorem so um i have a t well let's just read this first we have three sequences okay one sequence is a sub n this is smaller than the sequence b sub n or equal and the sequence the second one is smaller or equal than the third sequence okay for some number in on the on the domain on the x-axis and we also have the limit of the first sequence and the last sequence okay the limit of a sub n and the limit of c sub n as n goes to infinity equals l so what's happening here this limit and this limit on the top at the bottom is like a sandwich equals the same so that forces this limit of the middle sequence to equal l as well so the limit of b sub n as a sequence will be l and i have it written down in a t form if a sub n is less or equal than b sub n less or equal than c sub n for some number n zero and the limit of the left and the right or the bottom and the top are equal is the two limits are zero okay that means the limit in the middle has to be zero this is a picture okay just to show you that so a sub n as you can see has some is greater than b sub n and then the b sub n is smaller than c sub n so a sub n is smaller than b sub n i'm sorry i'm doing this backwards actually c sub n is smaller than b sub n and i have to change this sorry about that i have to change this to c sub n and a sub n and in the bottom of the yellow okay so um this is just to follow the um the rule that i just described so h sub n is smaller than b sub n b sub n is smaller than c sub n the limit of c sub n is l so c sub n is going to l as n goes to infinity and then the limit of a sub n is also in l as n is going to infinity so b sub n is dropped between these two that's what's called the square student so the limit of v sub n has to be l and for the example i'd like you to refer to your textbook so you can give you an example on this okay and we're almost done with the handout we have uh three more problems okay problem number ten what is a monotonic sequence and we have to give examples monotonic increasing and monotonic decreasing monotonic mono means one okay monochrome mono syllabic okay monotheist person who believes in one god okay monolingual person who only speaks one language this is mono one tonic is trained sound one sound one trend so the sequence could have one trend which is going up or increasing for some number in or it could be decreasing okay so there are two options for this for the sequence after some point after some n it could always be increasing forever or it could be decreasing so or neither okay so if it is increasing is monotonic decreasing if is if it's decreasing is monotonic decreasing and if it's neater means it goes up and down up and down right never that's the increase it never does a decrease in examples so i have two examples one is a sub n equals one over n is always decreasing how do i know this by observation one is the first term one over two one over three i see that is decreasing another way to find this out is to take the first derivative we're going to use calculus when i use calculus of the function associated in this case the function associated with this sequence is 1 over x so when i take the first derivative and we're going to observe what is the trend what is the behavior of this function so taking the derivative with respect to x is negative 1 over x squared this is always a negative number so for all x values of this function one over x including from one x equals one to x equals two to x equals 3 the function is decreasing because the first derivative is less than 0 that means that the sequence is also decreasing all right so we can do this by observation by using calculus next one i have one over um actually this is n over n plus one and over n plus one is always increasing okay how do we know that by observation we plug in a few values of n we get one half two thirds three fourths four fifths is increasing it's becoming bigger and bigger but using calculus we have the function associated which is x over x plus 1 which is the derivative when we find the derivative will be then and let's just fix this quickly this y here is not divisible so y and we get the derivative of the x which is 1 times x plus 1 minus x times derivative of the denominator which is 1 divided by x plus 1 quantity squared get 1 over x plus 1 quantity squared this is greater than 0 all the time for any for all x's including x's that are greater or equal to 1 the sequence goes from one to infinity right using the n values which are positive integers so the y is increasing that means that the sequence is also increasing okay so the sequence is also increasing let's move on number 11 bounded bounded so i'm asking you to give two examples one of each one example would be of actually two examples of bounded boundary sequences okay i did give you an example in part one the very beginning of this handout and then also into a but try to find different examples for number 11 please number 12 we have a theorem a theorem is a rule that can be proved every bounded monotonic sequence is convergent okay let's see if we understand that when i want to prove it every bounded monotonic sequence is converging the c bounded means that is in between two values monotonic means that is increasing other time or decreasing all the time after some number in let's go over this graph here this is a monotonic increasing function monotone is a bounded yes it's bounded by this limit this is ao is a bound because this function is not going to go over l and it's also bounded by the lower the lower value here this is a bound here this is one that's with a sub one so it's bounded by a sub 1 and l so these are the bounds since if since the function or the sequence is increasing what what is going to happen is going to increase and it can now go past the bound so it has to converge it has to converge okay so that's the whole point of this if a sequence is bounded and it's monotonic in this case monotonic increasing monotonic then is convergent for sure the other way is we have we have a sequence which is bounded okay it's bounded by this l value here but it's decreasing all the time after some n values and after some n value in this case i have one there that could be n sub zero this could be n sub zero but it's decreasing okay this is monotonic decreasing and it's bounded okay in this case will be bounded by these two terms a sub one and l so we're decreasing and is bounded it has to converge it has to come bridge okay there are exercises that i have that i like you to do on your own 23 20 seventy three seventy five and seventy nine these are in the book and section eleven point one please do them in order the videos that i have on the very top for you that i just said at the very beginning have the solutions to the videos i'm sorry to the exercises before you you can view the videos okay and then do the work but please try to do the work without looking at the videos too much and that is that is all for this section of 11.1