I. Tell. My. Home. Oh you bet. I think. I. Live. Oh you're low now to. $93.00 off wanted to end it this is the gear and reason Mitchell That's really funny we were in. Our mikes for a reason I would argue for like 5 minutes in our mikes weren't on no it's Ok it's only been a minute it's only been in it Ok well there's a repeat everything we just now can we restart Ok everybody that was listening on listen. Oh you're listening to $93.00 k. H.d.x. This is off on a tangent with Mackenzie Guerin and Reese Mitchell and Imus how we special of off on a tangent we're talking about some some tricky or not not really tricky but just funny and kind of goofy. Festive festive orders of numbers they're not they're not that festive but they're they're really fun I think they're really fun yeah so today we're talking about sequences and series 2 very closely related topics I just think it's funny because if you and I were like Why are Mike's weren't on we were talking about like we have like a tough start you know likable and behind and then like we realize we're not even talking you know I mean we're not even on air so this this is a funny show you know Yet it but it's going to be good I think I think it's in the it has had a lot of potential Yes sequences and series is the topic for today. And I was explaining to Mackenzie. Kind of before we were actually on air. But I can kind of do a recap but sequences and series. Oftentimes people don't learn much about what a series of numbers but they're a little bit more complicated than a sequence but sequences you learn a lot about. When you're younger maybe a 5th grade 63 even earlier than what you said it's all about having with some numbers and maybe there's a pattern when it really is just just that it's an ordered list of numbers so it starts with the lowest number and then there's a pattern say maybe they're all multiples of 3 and so that I just goes 136912 that's the sequence it's just a list of numbers that have sort of a common rule yeah it's like find the pattern or find them and so that like. Topic or that idea is on that you learn about early on and then you don't ever really do anything with it yeah it's just kind of I think it's just sort of to understand maybe the way that numbers are related to one another but then sort of in advanced math like in calculus. They come back they come back and you can use them. In a series form Yeah which is continuously add those numbers in a sequence together and think about if when you add an infinite number of numbers together if what that sum will be and sometimes equals infinity but other times the numbers in your sequence get smaller and smaller as the more numbers and sometimes they can converge to a finite value so these ideas that you learn about a long time ago they come back and more upper level math courses and they play a big role and these ideas that we talked about relating the calculus with limits and things we discussed last week just using a different perspective almost and that you're adding these numbers together smaller smaller Yes I mean I'm also thinking about last year I mean not last year last week when we talked about calculus the big. Theme of calculus is approximation so that's the idea that recess referencing when he's talking about limits is this idea that as you get closer and closer and closer to. A value or or in the in the context of a series when you just when you've added up so many numbers already and you're getting closer and closer and closer to a number at some point you can say no matter how many more numbers I add up it's going to be this number. That's really where the idea of like converging Yes and that's why that's why series are sort of a calculus idea or something that you think about in calculus even though it's not related to other ideas in calculus like rates of change and. You know sort of like the things that you can do with graphs you know they are intertwined ideas yeah yeah interesting stuff so we have a few examples of good sequences just to jump back you know definitely. Because because a sequence I guess the main distinction between a sequence and a series is that a sequence is just a list of numbers and they follow a pattern usually and then a series is when you some of those numbers together and sometimes you then subtract them you add one turn and subtract the next so there's different rules governing series all that fun stuff you can learn only calculus and cool new notations all that interesting stuff but for sequences we're just talking about the list of numbers themselves they were fun I guess couple of examples we can talk about one of my favorites 123456 yeah it's just the list of natural numbers right it's really just the only important thing to know about sequences is that the numbers follow a rule I guess this is 2 important things they follow rule and they go in increasing order that's all. So another example is 156-133-5129 yeah is there is there is there a rule for that one or did you I don't know I think that one was just. There and then I went down numbers to make the point and yeah it's an ordered list of numbers yeah. Yes I mean I'm sure there's there might be some sequences that don't have rule of rule yeah I mean maybe they're not interesting though yeah you probably I saw a sequence with a sequence probably Ok the definition of sequence doesn't include having a rule that governs which numbers are in the sequence but if you want to use a sequence to do anything you really want to have a rule because I want to apply it right because then if you can see if you can predict what the next number and you're like and if you can predict which number which numbers are and are not in your sequence then that is also echoing back to our function. Show yes functions give predictive power and that's why they can be so useful and so having a rule or like a function for the numbers in your sequence allows you to predict which numbers will and will not be in your sequence and that's why it's helpful that's interesting I haven't thought about that because really I guess you could think of a sequence as like a list of like your wife values from putting some x. Value Yeah you know and it's like alright add 3 to this number and keep doing it and when you go up one of the time with x. And use your new term and they are all that same pattern and sometimes they are not so easy to see I want to have 3456 that's just one to each new term right sometimes you might multiply by some number and then add a new number and then divide it and then it's hard to see those patterns sometimes but the ideas of functions kind of underpin that idea that a sequence follows like rule for determining the next number in. Yeah who stuff another interesting sequence that we like to investigate often it comes up in calculus a while and there's a lot of good physics applications as well it's called the harmonic sequence and when you look at the harmonic sequence it's very similar to 123456 but instead of having just the list of all the natural numbers you have the reciprocals most natural numbers so you go 11 over 2 and a reciprocal is just when you. So if you had like the number 2 the reciprocal would technically be 2 to the negative one exponent which is the fraction one over to you so there are separate 10 is one of the 10 Yeah I just mean a fraction and opening and upside down yeah and I guess another property is that when you multiply a number and it's reciprocal you get back one Yeah yeah so you know 5 times one 5th one Yeah good stuff so yeah the harmonic sequence takes the natural numbers and just looks at the reciprocals of those so $11.00 over $21.00 over $31.00 over 4 want to or 5 goes all the way down to infinity and. Some places that we see the harmonic series or a sequence is often when you talk about waves in physics and you talk about likes vibrating strings and those have a wavelength associated with them and the more like the higher frequency you like the more like wave weight you have within a certain and they go you know they go like one half. And that determines what they sound like yeah right there yeah harmonic Yeah there you go so the wave the wavelength follow the Harmonic Series yes and so. I remember that that's so interesting Yes So now just for all the people that are learning about sequences and series right now on the show the harmonic sequence last series. A harmonic sequence is just $11.00 half one 3rd one 4th and the series is particularly interesting because it's one plus one half plus one 3rd plus one 4th plus one 5th series and so you're adding them together and because this is sort of jumping to the calculus ideas but you don't need to know calculus to understand this is that when you add up numbers like this you're adding one and then you're adding one half which is smaller and you're adding one 3rd which is smaller and then you're adding one 4th which is smaller it feels like that should reach a certain number like you can add up all of these fractions but they can only get to a certain number as in the series will converge there say well this is the maximum that all these but all the reciprocals of real numbers can add up to it feels intuitive but what happens when you add up all the reciprocals you get infinity you get if it doesn't it doesn't settle down to of one finite number you just know that in a little or in like or not little or smaller and smaller amounts because you get so one over 1000000000 which is just a tiny little decimal Yeah you know but I mean really like I remember we talked about this series and the sequence in Calculus 2 and so many people had trouble understanding the series. So you won't find you. It's interesting because if you look at the graph of the harmonics. Series it falls like a log rhythm and it's like well like curve is is it doesn't have like it doesn't level off at any point it always keeps going it's going but it goes very slowly it's one of the slowest growing functions I think I think that it's really interesting to think about how uncomfortable it is for people to think about this series as being one that doesn't converge Yeah because it's growing at such a slow rate that it's really easy to say well it just it just up at some point it approaches something but it doesn't it's always growing even if it's a teeny tiny little bit which can reinforce I guess the idea that there are series where when that you really do approach a value and at some point. You just get closer and closer to that value and you never go over it and that's what it means to converge and so the idea that this sequence or the series doesn't converge. Sort of like I don't know says to me like. Like the mathematical theory is that even if it's growing like one over a 1000000000 like it still growing Yeah you know and so that just makes the idea of convergence like you can add an add an add an add an add but no matter what it's not going to be over a certain number for other series it's like an extra interesting Yeah because we talked about calculus with Dr Campbell back throw back and that's why I want to also an aside a tangent tangent. That was one of my favorite classes of college so far and it was so difficult at times it was a lot of it but it was so neat learning about the series and sequences and seeing how they really relate good stuff but Dr Campbell told us that comparing the harmonic series to other series can be a good indicator. If that other series is convergent or divergent so you can look at. The harmonic series and you can just kind of generalize it to one over in being like that and number so like so you're at the 30th you're in your c. Series it would be one over 30 you know. And you can look at other series and if they. If the values of that the other sequence get smaller faster than the harm does that make sense yes like if the numbers get smaller and smaller even quicker than the harmonic series that might be a good indicator that that series converges and virtuous if they get smaller at a slower rate than the harmonic sequence or series than they probably diverged as well so you can kind of use. The harmonic sequence as like a ceiling or like a benchmark for examining other sequences and series which I thought was so interesting you can like because it's not so much just like. Calculating It's like a very comparative way of looking at this is in series and it's it's not intuitive often but it's a very unique and interesting application of the study you have like all these different tests that you perform on sequences and series it's just it's so interesting how to me like knowing if a series is or is not going to converge it is or it is not going to add up to a number even if you add up infinite number of numbers in that series for this is a lot of like adding in numbers and I'm sure this is hard to listen to and understand but. It's not intuitive to know what will converge and what won't and it is just like how weird is it that you can add up an infinite number of things and no matter what they always equal the same thing it's really interesting and sometimes it doesn't sometimes it doesn't and sometimes it looks like it does but it doesn't Yeah I think I don't I think the harmonic sequence is very unique for that reason particularly. Reste you want to introduce the funny sequence that we have Yeah this one this sequence is really really good yet this one is kind of like our this is serving as our math joke of the show because it's very like kind of goofy It's tongue in cheek it's tongue in cheek yes in a way this sequence is called the look and say sequence and it was. I guess conceived right I guess like it was made up by this mathematician named John Conway and I think he's still alive today he's a fairly contemporary not one of those historical figures we like to talk about on the off on a tangent. Time he's no with accuracy. He's not as a Little did come up with something really cool you know you hear just like I know he's for he's a famous mathematician Yeah sure but his sequence is what the look and say sequence so much fun the 1st term you start off as one and then to generate the next term in that sequence you look at your previous term and you say how many of that number you have so I would look at the number one and I say oh I have 11. And so then the next term is 11 so I said I leaven Yeah and says Oh look I have 11 in my 1st term so no 11 becomes my 2nd term and so for the 3rd term I look back to my 2nd and I say Ok I have to wonder now and then my 3rd term is 21 and you just keep repeating that process yeah and so for the 4th term then you'd say for 21 well I have 12111211 Yeah and it just keeps going and going and going and going and going and there's a lot of 2 someones and 3 is they get thrown in there yeah it's so much fun but I can't remember if there is a very specific application that this look and see if it has other than being like phonetically and numerically fun yeah yeah I'm sure I'm sure there's fights on like weird. Application forwards but yeah the look and say sequence it's not it doesn't follow any sort of arithmetic or adding and subtracting and. You know multiplying dividing anything like that. It just simply uses the fact that we use the English language to describe numbers verbally that that process is the rule for making the next turn which means that you can make you can make a sequence however you would like however your heart desires Yeah I have one that can be like a homework assignment like everyone listen think of a funny sequence that you want to make and and it will be your sequence Yeah just start you can and you don't even have to start off with a one you know oftentimes these a lot of times they do start off with one but I mean you could start off with 10 or 15 or anything in any and then you do Infinity plus a. Whatever and then it's going. Sort of out there yeah I noticed I didn't say 7 because oftentimes feel like oh it's just say you know I mean like that's you know it's like or 3 given I killed people or think of a number and so they're $7.00 and $3.00 that's what it's like for me yeah. 82735 and any number you want to any number and you just apply a repeating rule to it and you can make your own sequence so much fun so much fun. We are going to have a song break sort of so that everyone can i death digest sequences and series so that we can talk more about them about famous examples or yeah ways in which people actually use these ideas to do math and not just have fun with them even though they are really fun. Ok Well our song is threes and sevens which is so funny Yeah because we're just thinking about 3 isn't. There even though there's a song is where you trying to cue up the song Reese No no no no I was saying that people like when you say hey think the numbers we want to turn is like 7 you know like that people go to that one more often than others but the other song is the reason sevens by Queens at the start this is perfect we were planning on playing the song last week but we just don't get to it and it's so good it's like it's like the world is telling us that it wasn't correctly time to last week and now it is perfectly timed this week so enjoy this is 3000 seventh's by Queens Of The Stone Age. . And we're back you're listening to off on a tangent with Reese Mitchell And we can see this is Katie x. F.m. Conway Arkansas we were talking a little bit about sequences and series prior to our song Break discussing some famous ones ones that have unique physics applications ones that have we are property of the hard ones that mathematicians have been thinking about for a long time yeah I don't I don't I forgot to mention this but the harmonic sequence I did I did I did a paper about it Dr Campbell also throughout the calculus we got to pick a topic and just write about it and I wrote about Bernoulli like proving that it was divergent and I was in with like the 720 s. Yeah yeah this is more like it's just sequences and series are really fun for mathematicians to think about and it's like because it's not just computational It's like theoretical and computational if you really know itself like the nature of you know numbers and like how we try things and like and like the ways in which numbers like behave. That are predictable so that seems to me like a way that math is. Like sort of something we're discovering you know like we did make up numbers in a sense to be able to count but also they're so good at explaining the world they are so they do things that we didn't. Well you know why we also did when we were like making up number theory we weren't like Ok now we want sometimes when you add numbers for them to add up to one thing and sometimes we want them to not add up to one thing like that wasn't something that we like that wasn't a goal of number theory but it is a property of numbers so I mean I think that whole field of like number 3 and so interesting numbers are uncontrollable Yeah even though we kind of created them yeah like certain numbers or even huge ones they have these really unique properties that are like oh yeah a few other ones that we know about also are unique prime numbers or yeah I don't know it's like the distribution of prime numbers this year yeah that's like a huge topic is like you know a rule for that there is because there's also so this is a tangent and it can we do it can turn into go for it interesting about prime numbers. But there's a conjecture called the 20 prime conjecture and. But you have numbers like 3 and 5 that are always like one apart from each other 3 you go up 152 numbers 30 called Twin Primes there's a conjecture that there is an infinite number of 20 primes no matter how far out you get how big your numbers get you always have when primes at least somewhere you get a unique Yeah I'm just trying to make sure that. Oh my gosh this is