❖ Root Test for Series ❖

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In this video I look describe the results of the root test for series which helps us decide whether or not a series converges absolutely (absolutely convergent). Note that if it converges absolutely, then the original series also converges. (Note the root test puts the terms of the sequence in absolute value so it could change signs around).
The root test involves taking a limit. Note that if the limit L is greater than 1 (including infinity) then the series diverges.
If the limit is less than 1 then the series converges absolutely.
If the limit equals one, L = 1, then the test is inconclusive.

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16 сен 2024

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Комментарии : 336

@patrickjmt 2 дня назад

Hi all! Wanna help a RU-vid education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

@pranavranganath7192 3 года назад

Almost 13 years later.... still amazing. Thank you so much for this. It's taught timelessly.

@girmawakeyo4143 Год назад

yes

@Salamanca-joro 5 месяцев назад

Damn man i was 3 years old when he uploaded this video, and now look at me watching this video after all these years! RU-vid is great

@Merchantic 10 лет назад

Still depressed that I've learned more from an hour of your videos than a month of class.....

@MarcoMorelos 10 лет назад

well for one thing, He doesn't go over proofs, which are extremely abstracted. He just tells you how to do the concrete part of math: solving. The simplicity of the way he teaches plus the fact that the videos are short and convinient makes it easy to learn here. The people coming here are also in the mindset of learning too, otherwise it would be unlikely that they'd come here at all!

@patrickjmt 15 лет назад

glad you like them all : ) i figured that if they suck, i would just delete them. most people seem to like them though : ) and honestly, i forget half of this stuff, so i make them for 'future me' - the one that has forgotten and re-watches old me (newer me?), so i have to be nice and soothing to my future self : )

@josephabbate6315 10 лет назад

Dude, I was literally tweaking out about this midterm that I have to take in like 40 minutes and your videos on convergence tests totally saved my ass. Thank you, Patrick.

@lukemattison7836 10 лет назад

6:11 "The numerator is going to get big very quickly, the numerator is going to get big as well, but not as fast as the numerator." Idk why I found that so funny, probably too much studying...

@KD35USA 11 лет назад

"Tired of scrounging youtube for math help?" No, I've found this channel. Why would anyone pay for online tutoring videos if they've found patrickJMT?

@sashamuller9743 4 года назад

7 years down the line and us bloaks still get that stupid ad lol

@dennyyang7286 10 лет назад

For the 3rd question he did, lim n->infinity (1+(1/n))^n is in fact equal to e. Set y=(1+(1/n))^n, then take the natural log of both sides. You will have ln y =n*ln(1+(1/n)) Then take the limit on both sides. lim n->infinity lny = lim n-> infinity (n * ln(1+(1/n)) Now, the goal is to rewrite it in an indeterminate form in which the L'Hopital Rule can be applied easily. lim n-> infinity [ln(1+(1/n)] / [1/n] which = 0/0. Take the derivative on the numerator and denominator and you will be left with lim n-> infinity (1/ (1+(1/n))) and that limit will equal to 1. Now we have lim n->infinity ln y = 1 lim n->infinity e^(lny)=e^1 and then replace y with y=(1+(1/n))^n after you cancel out e and the natural log. Therefore lim n->infinity (1+(1/n))^n = e lim n->infinity (1-(1/n))^n is equal to 1/e.

@himanskss3749 5 лет назад

thank you

@patrickjmt 16 лет назад

well, it is basically knowing that the n/(n+1) is the flip of (n+1)/n; that raised to a power of n has something to do with 'e'! i am seen that limit so many times now, that i just recognize it. so to answer your question, what made me think to do that is: experience! sometimes one has to be shown what to do a few times before it sticks!

@yasthilmaharaj 15 лет назад

"most people seem to like them though" No No Patrick. Everybody LOVES them. You are a great man. Thousands of students around the world are more grateful than you know. Thank you for the time and effort you put into these videos. They have saved the lives of many students.

@datboytroy9393 8 лет назад

I would be so lost without you! I can honestly say I learned more over the course of watching your videos in one day than I learned the past 3 weeks in my Cal 2 class!

@aoalr 8 лет назад

You're literally my savior. I have an exam today and my professor was crap at explaining everything, so I didn't understand any of the series and sequences. I watched at least 8 of your videos and they helped a lot! Thank you so much, Patrick!!

@patrickjmt 16 лет назад

series are really cool. it is one of my favorites subjects (if not my favorite subject) that one encounters in calculus. some very beautiful and useful results come from them. of course, they are also pretty tricky at times!

@patrickjmt 16 лет назад

1 to any real number power is 1. the problem is if you have a function of the form [f(x)]^[g(x)] and f(x) is APPROACHING 1 (but not necessarily equal to 1). limit notation is not very good; the standard notation is to use equality, when in reality it is just 'getting close' to that number (in most cases). eventually, one figures this out and just learns to deal with it : )

@patrickjmt 13 лет назад

@remirap no, i am long done with school.

@vjbhatt96 13 лет назад

these videos are very much helpful i have been watching sequence and series topic for almost 8 hrs with small intervals and practicing problems in between ....i hope i will do well in exams thank you so much ....people like you should always be there ....!!

@16colombiano 8 лет назад

you know you made it when you say "The answer is the number e"

@dylansanderson3663 7 лет назад

e is a number though

@chelseasrenee 7 лет назад

Yeah, but stating e as the answer is more exact than saying 2.71 is the answer. For example, if for a problem you find the answer to be sqrt 2, it is much more accurate and precise to let the answer remain sqrt 2, even if you have been taught to simplify. This is because saying sqrt 2 = 1.41 is actually incorrect, even though sqrt 2 is approximately 1.41 if one is using three significant figures. So, you could indeed say the answer is approximately 1.41 (like if you are doing a physics problem where estimation>accuracy), and you would be technically correct, but it is much better to say sqrt 2. There is just no point in simplifying, it only takes more time to produce a crappier answer. So, like +poopnuggets (lol) said: to answer e is to be exact, to name some number of digits of e is the answer is less exact. It is unnecessary to simplify, much like it is unnecessary to rationalize-- just more time and work for a less correct answer. I do not know how understanding an answer can be e is equivalent to "making it," but realizing that a concept like e is more exact than its estimated number is indeed helpful in math. Because, actually, e is not a number but a constant that can be expressed precisely as the base of the natural log (ln), such that ln(e)=1 (e^1=e), and can be estimated as the infinite series of 1/n!, which does not have an exact value since infinity is not a number either, just another concept, but does give us the estimation often used (2.71).

@chelseasrenee 7 лет назад

You can also do a lot of cool stuff with e that you cannot do with anything else (like its role in probability theory), and it has unique properties (such as the derivative of e^x is e^x, and the same applies for riemann integrals). ALSO, all digits of the constant e are not known (it is irrational), though the known number of digits is constantly increasing

@lukeeaston7231 8 лет назад

6:12 "the numerator is going to get big quickly, the numerator is going to get big as well, but not as quick at the numerator"

@paulabdo1428 8 лет назад

lmao

@FcBarcelonaKid 8 лет назад

lol

@Afroninja10000 7 лет назад

I'm looking like a dumbass laughing in the middle of the library right now...

@FlakeyOkie64 12 лет назад

just started studying this in calc II and your video was very helpful. sound quality was good and you made this topic less frightening!

@sasaltthefries 16 лет назад

I love the videos - they're so clear and really add to what I try to learn in class (we just go too quickly!) Thanks for doing this!

@patrickjmt 16 лет назад

but only the 1/n term goes to zero so that we are left with (3^0)(3^3) = 1(27) = 27

@patrickjmt 13 лет назад

@BlueColourPencils awww, thanks! go texas!

@patrickjmt 16 лет назад

i need to start paying you for all the nice ratings and comments : )

@rahlity 13 лет назад

God bless you dude! You have no idea how much these videos have helped me for my cal 2 final. the limit of my appreciation for your work approaches infinity. Great work!

@jesscap09 11 лет назад

I spent hours trying to understand my professor's instructions... yet I watched 3 of your videos, and I'm good to go. I've been barely passing Calc II, and not because I'm "dumb"... it's because I've spent an insane amount of time trying to understand HIS teaching. I should have sought other sources, and I would've done so much better in the class. I'm studying to be a teacher, and this really makes me see how there is NO "dumb" student... different students learn differently.

@MMAWatch7 7 лет назад

Another few videos to once again get ready for yet another Calc 2 test. Thx Patrick you tht man.

@patrickjmt 7 лет назад

good luck!

@neggsa 11 лет назад

Patrick I love you! Your videos have helped me so much throughout calc 1 and now calc 2! Thank you for making these videos, so many of us appreciate it!!

@patrickjmt 16 лет назад

i think you need to flip your fraction (n+1)/n

@patrickjmt 16 лет назад

ops, thanks! i did that in another video too : ) i am making sure everyone is paying attention!

@bastianian2939 4 года назад

So good man. Still helps people out after 11 years

@patrickjmt 16 лет назад

glad to help it make sense for you!

@ThangkhungNay 14 лет назад

Geez you make it much easier to understand than my professor...great stuff!!

@beansombrero 14 лет назад

@patrickJMT no theres a place called jordan located in the middle east. You are making such a huge a positive feedback

@ShaneBurroughs5 13 лет назад

just so you know your the man. . .helped me through my calc II class with relative ease. Best vids on the net for sure.

@patrickjmt 16 лет назад

my pleasure my friend! hope the exam turns out well

@raydredX 12 лет назад

It's correct because: 1=A/A for any A so: 1=(1/n)/(1/n) He just multiplied it by 1 so it doesn't change the result. Example: 2/3 = (2/3) * [(1/2) / (1/2)]= (2*(1/2)) / (3*(1/2)) = (1/(3/2)) Ignore the example if it's confusing. He multiplied by 1. That's just it.

@DoggoWillink 12 лет назад

I thought series were going to be hard, but this whole concept so far, including doing differential eqtns with power series, and the Taylor polynomial section, has been easier than integration techniques IMO. First time in a while that I didn't actually need Patrick's videos to get it down correctly. Still watched them though, they always have something inventive to teach you.

@vjbhatt96 13 лет назад

thank you so much for being there i have been watching you videos for the whole day today about this topic..........they are very much easy to understand..thank you again!!

@patrickjmt 16 лет назад

thanks again my friend!!

@sindresaetre 16 лет назад

Great videos, they are really helping my preps for the exams! By the way, at 5.45 you say that 0*27=27, not that it matters, the answer will still be the same.

@mitmfan 16 лет назад

You did an excellent job! I only wish that I had discovered this while I was learning about this instead of after I had already finished the course. However, I never really did understand it during the course, so I was lost during the assignment and exams, so this was like me learning it for the first time. However, my calculus teacher seemed to avoid doing examples or at least examples using numbers like you used. Again, good job! I know others will benefit from it like I did.

@patrickjmt 13 лет назад

@Raxarax i was being a bit ' tongue in cheek ' with that comment

@patrickjmt 14 лет назад

@BornAtTheBar yep, got a masters

@patrickjmt 14 лет назад

@5ANASHEEROA do you mean michael jordan? was he in the middle east?

@alexatlanta1983 12 лет назад

good way to remeber the tests (R.eally R.eally A.mazing D.octor In C.anada) D: Divergence Test A: Alternating Series Test R: Ratio Test R: Root Test I: Integral Test C: Comparison Test

@patrickjmt 16 лет назад

no problem

@dolce0me 15 лет назад

OMG I agree with Inferfire........it is plain scary that I understand all your root test/ratio test stuff, I have my final tomorrow and before these videos I thought I was screwed because i have absolutely NO idea what my prof is babbling about....I wanna seriously thank you so much!!!

@Abraham_Justice 14 лет назад

For the last example on this video it is easier to use L'Hopitals on that limit by taking the ln of the limit instead of having to memorize that that limit equals e. You can show why it equals e.

@maethorlotr 14 лет назад

@patrickJMT Thanks for all your videos!! Its much easier for me to study math when I can pause and rewind the lecture :) But I have a question, say I have a sum of (x/n)^n what do I do with x? at the end I got: lim (x/n) do I say that it converges for x

@vjbhatt96 13 лет назад

@patrickJMT i like it when you go into the conceptual depths but at the same time keeping maths enjoyable, i feel really grateful to you., you just inspire me.....i like "improper integrals" in your video series very much :-)

@sebastianzx6r 11 лет назад

Are you talking about the part: n/(3^1/n *3^3)? If you look at 3^1/n as n goes to infinity, it becomes 3^(1/infinity). That equals 3^0, remember this equals 1. Your denominator just becomes 1*27

@tonewardbound 8 лет назад

I like how at 6:12 you referred to the numerator twice just to test if folks are paying attention. ;)

@sebastianzx6r 11 лет назад

Ratio test is usually when you have factorials and exponential stuff. Yes all this stuff is raised to the n^2, but if you threw in n+1 for all the n's, simplifying it would be a nightmare. Whenever you see an entire series raised to n, it's best to use the root test because you will eliminate the n or reduce it's power like in the example this dude did.

@AioTranNhan 10 лет назад

(1 + 1/n)^n = e ? i dont get that part and also, why dont we apply lopital rule at 8:40 ?

@patrickjmt 10 лет назад

you can show the the limit as n->infinity of (1 + 1/n)^n = e by using l'hopitals rule. you dont use lhopital at 840 since it is not an indeterminate form; it is 1/e.

@AioTranNhan 10 лет назад

patrickJMT thank you patrick, you're the best, I did take my quiz today and it was fantastic. Thank you so much

@patrickjmt 10 лет назад

my pleasure! glad to hear the quiz went well!

@marquez2390 6 лет назад

Your last example really helped me as I wondered if we would have to use the root test twice.

@zchosenyoutuber3675 8 лет назад

isn't 1^(n) as the limit goes to infinity and indeterminate form, therefore applying L'H rule on 8:40? It is to my understanding that 1^infinity can either be 1, infinity, or undefined, but how do we decide on the result ? Should i always assume 1^infinity is one?? My head hurts..

@mikailconstantbilyamin1832 7 лет назад

zChosenRU-vidr 1^infinity = e

@fkncompton7124 5 лет назад

Mikail Constant Bilyamin I don’t think so lol I think it’s just 1

@IkeSan 5 лет назад

I love how I got a video saying that videos from the 2000s math are not educational enough when this was a really good video.

@FcBarcelonaKid 8 лет назад

God bless you Patrick

@heytarotpipi 14 лет назад

omg, thanks a ton patrick!! i have a maths exam next week, luv ur videos so much

@smatrick0208 11 лет назад

you sound like the hippie teacher on beavis and butthead. On a serious note your videos are very helpful. Thank you for posting!

@patrickjmt 15 лет назад

ahahahhaahah - that is awesome! kisses + cookies = a good day

@AnthonyPickett 12 лет назад

wow. im at UT too.. and the whole time I wondered where patrick was at and it was right here.. same world!! Just show Austin rules!

@FreeHonesty 11 лет назад

It's the exponent for 3^(1/n) that equals 0. As in you will get 3^0 which is equal to 1.

@concisestem 13 лет назад

I think the second example has a small error. The denominator has the (3^3)(3^1/n). You say that this becomes 27 as n goes to infinity. I'm pretty sure that it becomes (9)(1) not (9)(3) since 3^0 is 1 not 3. Doesn't matter because the numerator is infinity, but still a comment is warranted.

@Closing408 12 лет назад

due you mean why he raised it to the (1/n)? and if thats the question its because the equation has you find the limit of [an] to the nth root which is the same as raising the problem to (1/n). an example is the the square root of 4 which can be written as 4^(1/2). hope this helps :)

@mitmfan 16 лет назад

I understand that if you have a function raised to the nth power, it is a good idea to use the root test. However, what about the other tests: integral test, sequence test, comparison test, and ratio test? What should you look for in a function to determine which of the other tests you should use?

@seanmartin7257 5 лет назад

I know this is an over ten year old video but Patrick I hope you're having a good day

@MIDNightPT4 5 лет назад

Sean Martin lmfao

@inside91 9 лет назад

I'm not sure if this is an active conversation but if someone sees this please respond. In the proof of the Root Test I do not understand why you have to choose a q, Q

@joeydaboss1001 9 лет назад

I'm confused at 7:52, why did he move (1/n) onto both the numerator and denominator after distributing it already??

@forevereveryours 9 лет назад

JocoFitness He is doing a clever form of "one"

@Romans--pe4yh 5 лет назад

Rule number one: never and I mean NEVER question Patrick

@vdubnthehouse 15 лет назад

Very helpful. Much better than my cal II teacher who assumes we know everything.

@DoggoWillink 12 лет назад

@kenikozo Oh sorry, you said limit, not series. I don't know then. Because if you have a variable in a basic limit going to infinity, that variable is what would be replaced with infinitely increasing numbers. The only thing I could think of is a constant to the x power, like 2^x, where x approaches inf. Though, I probably am forgetting something. Anything to the "infinite power" would be increasing or decreasing without bound though I'm pretty sure.

@hash2pat 15 лет назад

dude ur a lifesaver. Thank you for all this. You make so much easier

@emjab68 13 лет назад

Thank you so much for making this easy to understand! Fantastic.

@jiamonx2 13 лет назад

@patrickJMT i mentioned the same thing on one of your other videos also, i think the reason why people think it is boring in a classroom is because computer are just fun, or some kind of drug that emits out of our computers, lol

@BreakneckWalrus 12 лет назад

Great video! I have a test on Tuesday and your videos have been so helpful. I'd just like to let you know, though, that this video's not on your website. Thanks again!

@concisestem 13 лет назад

@js6781535 No, it looks like it will at first glance but the number that is being raised to the n'th power is approaching 1 as n goes to infinity. ie - (1+Very Small Number)^(Very Large Number). As he points out, this is the definition of the number e or approximately 2.718 ...or 2.7 1828 1828 45 90 45 (broken up into the memorization pattern to the 16th significant figure, like anyone would ever do that)... Still absolutely convergent either way, but 1/e was correct.

@199alexman 11 лет назад

first i would like to say you have nice picture, second is so if you change n in terms of X then divide all numerator and denominator by x then you know the rules from there in top u will get x/x which is one down you divide 3/x which is zero time 3^1/x is zero so u will get 1/0 !

@MrMankoe 13 лет назад

Wow! incredible... especially the very last part of the root test!

@omnifur 14 лет назад

YOU ROCK. I have a calc test this upcoming monday

@belacile 2 года назад

6:08 [The second time he says "numerator," he means "denominator," but I still find this bit very amusing] "If you start plugging in large values for 'n,' the *numerator* is gonna get big very quickly; the *numerator* is gonna get big as well but not as fast as the *numerator.*

@patrickjmt 15 лет назад

well, 'e' is used in the continuous compounding formula. and dont go hatin' on poor ole ' e ', it just wants to be useful like all the other numbers!

@MMX0710 15 лет назад

Normally you can use both ratio and root for these questions. it best to use root test for questions like (n)^n

@emirdumani 15 лет назад

your videos are brilliant patrick, just brilliant :) by the way, i wanted to ask if n-th term theorem could be used on your 3rd example here.

@TazzyZee14 10 лет назад

English… sweet, sweet english… TT^TT I can actually understand what you're saying. Please… come be my professor!!!!!

@abdullaha9586 6 лет назад

How did you get 1/e? Confused on that part, unfortunately.

@Gunz_o 5 лет назад

lim (1+1/n)^n=e

@armidylano44 13 лет назад

@patrickJMT Hey, I noticed you're from Austin. I'm actually at UT right now =) Anyway, thanks again. King of youtube math, you are.

@kyosukeplays 11 лет назад

I love seeing all of the steps. Thanks!

@1mohiuddin 12 лет назад

Patrick , could you explain why , with root test in example 3 ( 1+ 1/n) ^n=e . I really forgot what is this e means. Thank you.

@patrickjmt 16 лет назад

how about: THANKS PATRICK and youtube... ? : )

@estebanquintanilla6959 5 лет назад

Bruh. If I pass my Calculus 2 exam. I’m so sending some cash your way and making a donation. You definitely deserve it! I’ve been using your videos since pre Cal. I don’t know where I’d be if it wasn’t for your videos. Thank you so much!😭

@love-hammer 10 лет назад

at 5:47 why does that equal inf/27? since 0*27 = 0 then wouldn't it be inf/0(another thing I don't understand)? I guess I just don't know what limit law makes that valid.

@love-hammer 9 лет назад

damn dude I was just asking a question. EDIT: ignore this. some asshat was being an asshat and his comment got deleted.

@normanchang3911 9 лет назад

broadswordslayer lim(n→inf) x^(1/n) should be 1, x for any real number you can use a calculator to try it.

@DoggoWillink 12 лет назад

@kenikozo Like an "n" in exponent right? That is a geometric series. I don't know if "non-geometric" series can have an exponent of n, maybe it's possible; but they'll likely always be geometric, ^n. I'm sure Patrick would correct me if I'm wrong.

@n3vrax 15 лет назад

It was not a mistake...it is 3^(1/n) as n->infinity so it would be 3^0 which is 1. It doesnt matter anyway...it was just to make it clear. Thanks listening to me :)

@jesscap09 11 лет назад

...and btw... THANK YOU. Your instruction is fantastic. I'm so grateful to have found these videos.

@flawns 13 лет назад

dude i haven't even reached at this level of math yet , and im just sitting here and watching it....idky but who knows maybe it will help me out in the future

@MrBlash93 10 лет назад

Thank you so much for all your videos . Your videos have truely helped me.

@Voytik4 10 лет назад

Doesn't 3^0 equal 1?

@szoutenb 16 лет назад

Thank you so much! I've been having trouble with Series! It'll certainly help my final today!!

@atifchaudhry1183 11 лет назад

For the end of 2nd problem, where he says (n/(3^(1/n) * 3^3)), he is referring to 1/n going to 0, which would mean (n/3^0 * 27) = n/1*27 = n/27 which is inf/27 = infinity! And to know where he got the 'e' in the 3rd problem, read this Yahoo answer: answers.yahoo.com/question/index?qid=20080510201020AA5K1x8