Power Series - Finding the Interval of Convergence

[STUDYLATER1]

Hi Patrick! I have a question. What if the x function is squared e.g. -1<[(x-5)^2]<1. So how to you find the range of X

Thanks by the way your videos are just awesome. Watch your videos is so much better than sitting in lectures which i dont understand.

[jkomo15]

on the second one, could you have done root test instead of ratio? thanks

[patrickJMT]

@jkomo15 try it and you can tell me

[chlejrdud3]

Fucking. Awesome.

[misterthom100]

legend!

[andyplace]

You write all the same steps as my calculus 2 teacher. The difference is, you actually make sense and explain what's going on at a perfect pace!

[Pajameister]

Thank you, thank you, thank you! You never fail me, I couldn't find any of the convergence tests on Khan Academy, but it's patrickJMT to the rescue!

[patrickJMT]

@Pajameister tons of sequence and series (and other calculus stuff) here! come back any time

[thepinkbed]

I do have a question about after you took the absolute value of x out of the limit...do we need L'hopital's rule for showing that the limit with the 'n's inside is 1 since it IS infinity/infinity? Thanks! (I'm eating my lunch while watching PatrickJMT calculus videos so I don't bomb my test in the next couple of days...)

[Limabean231]

@thepinkbed You can use L'hopital's rule to prove that the limit is 1, but you can also multiply both the numerator and denominator by 1/n (essentially multiplying by 1). This leaves you with radical (1/(1+1/n)) which approaches 1 as n approaches infinity

[bedukfan]

Thanks

[tiff4annie]

Hey Patrick, do you have any videos on finding the formula for the sum of n terms or finding the formula to find the limit as n->infinity

[ThePositivePaul]

you are getting me through first year University Calculus

[Almighty256]

@patrickJMT I'm confused... I thought that if the result of the ratio test was a limit of infinity, then the series was divergent.

Aren't we looking for the interval on which the limit of the absolute value is < 1?

A limit which is < 1, but is not dependent on x, should indicate convergence along the entire range of real numbers. But Willy's limit indicates divergence, does it not?

[patrickJMT]

@Almighty256 ops yes, zero : )

[TheWillyDix]

If I found my limit to be infinity, what does that mean about the interval of convergence?

[bossbower]

@TheWillyDix no, if the limit is infinity, then the interval of convergence is {a}.

[patrickJMT]

@TheWillyDix it is a single point

[bingavtoski]

people should appreciate you more

[runeruly]

no one in my class pay attention to our prof. b/c unlike our prof. we can rewind you over and over and u r great at explaining.

[dbramm01]

You are a godsend!! I should be paying you instead of Drexel to teach me math.

[runeruly]

thanks a lot bro

[Rudy727]

omg.. i didnt know this existed.. i wouldve passed my test today.. fuuuuuuuuuuuuuuuck.. i missed the lecture in class and i dont have teh book..

[HmmmYesQuite]

Why oh why have I not discovered these videos until now? I have a feeling that calc. 2 would have been a lot easier.

[thearunnagar76]

screw our teachers they dnt knw hw to teach u hv a pleasant voice tooo ........ wud u mind 2 rotate ur camera 180deg so dat v can see our guru :))

[Jaquis198]

you made calculus possible for me haha thank you SO MUCH!

[fishingchamp77]

you... are..... AWESOME!!!!!

[MsPowerade123]

great video. thanks for the help

[kryllpayne]

After an hour of being stuck on these damn intervals of convergence, you have saved my night!

[HamletSlaya101]

Hey Pat do you have an example where the limit diverges like instead of @6:33 it converges to one??? Please and thank you

[Xetian91]

Thanks so much for the clear video! Had a family emergency and missed all this crap but you cleared it up far easier than my book.

[jaycenicole]

what do you do if when you get to the point where you take |x| to the front, your limit equals infinity?

[tajuara100]

You are my hero!!!!

[09snaider]

In 2:00 why does the limit equal 1 ???

[robertobiero]

you are godsend men, got a 98% in my calc exam because of you :)..thank you soo much

[fjksalf]

you skipped a few steps, like how you need to use l'hopitals rule to properly find the limits, otherwise you have infinity over infinity. for the second test, the root test would be easier. and you should have done an example where the limit doesn't equal 1.

[Breezyy527]

@fjksalf

Donate some Ca$h and he will surely do it your way.... or maybe not..

[patrickJMT]

@fjksalf feel free to make your own far superior videos instead of trolling mine.

[oli696969]

@patrickJMT lol he was giving constructive criticism, how rude and ungrateful can some be?

[mhlokotsi03]

you don't even need to use l'hopital's rule, you can just divide through with n. STOP BEING SUCH A TROLL!

[McBirdec]

there's nothing better than pressing pause during a lecture

[emredodii]

Great stuff! I didn't know how to interpret the result after the ratio test, but now I do. Thanks!

[teshsherpa]

What about using root test ? or is it that we can only use ratio test ?

#Your videos are helping me a big deal.Thanx!!

[powercrazy11]

@teshsherpa the root test is a basically only useful when A sub n is risen to a power of n so that when you take root n of it, the function becomes simpler. TL;DR root test is a different way of finding out the same thing that the ratio test finds but in more refined cases.

[06439863632]

also i noticed that you pulled the x - 5 out. does that mean that x is a constant?

[blaquestardust]

@06439863632 (x-5) can be pulled out of the limit because the limit doesn't depend on x, only on n.

[06439863632]

i got confused at 4:05 does that mean that at -1 it's convergent and 1 is divergent?

[mrhozer]

@06439863632 Yes, when you take the limit of the series when x = -1 the series converges to some finite point. But when you take the limit of the series with x = 1 then it diverges to infinity.

So overall, the series is convergent for any value of x from -1 (including -1) through up to but NOT including 1.

This is written as [-1,1), or -1<=x<1.

[-1,1) means all the values between the two numbers, and the bracket [ means including, and ) means not including.

[canUfeelthaheat]

Very helpful. Thank you very much.

Just curious, what kind of education do you have? I'm guessing Master's/PhD in Mathematics of some sort?

[patrickJMT]

@canUfeelthaheat MA in mathematics

[darthmath2]

@patrickJMT Don't you just love power series? I am undergraduate level and I just love power series. I never found something so beautiful. I can't wait to further explores this at a graduate level!

[patrickJMT]

@darthmath2 i do like series, some of my favorite stuff

[patrickJMT]

@patrickJMT from calculus, that is

[GG4GlamourGalore]

You are seriously the biggest LIFESAVER there is. All my fellow engineering classmates agree with me on this. Keep up the good work and thanks again!

[patrickJMT]

@GG4GlamourGalore glad i was able to help some of you : )

[slat13]

I was wondering: Do you always have to use the ratio test to find the interval of convergence of a power serie or can you use the root test for example?

[TheDancingPen]

SUPER! :D Thanks and keep uploading such videos!

[fishzard]

Thanks Patrick, love your vids :)

[SpnFan04]

wow. you are amazing. THANK YOU!

[NeetFish]

Thank you

[revolution09pac]

you rock.

[xXDoraThaExploraXx]

who needs college when there's youtube

[watermelonandpotato]

I find it quite funny that the view rate spikes about four times a year, or twice a semester!

[danthemantmac]

Thank you so much for taking time to put together such a good video, this helped me so much, i appreciate it

[foosta5]

whats the radius of convergence for the second example

[TheNassrawi]

my final exam is after two days from now ... & you helped me alot

Thank you .... my teacher

[AllDayRippa]

why is there always shirtless guys in the featured videos????

[TehFranci]

you helped me pass Calculus. ha.

[liangsupree]

OMG!! Thanks a lot again !! I thought I'd fail my tomorrow's calculusII exam but now I feel much better. BIG THANKS :))

[noyan000]

I've been watching your videos for the past two days. Time to take my calc 2 final >: )

[Xeribol8291]

This helped me SO much! I understand this stuff so much better coming from you than I do from my professor and textbook. Your videos will definitely be used as study materials for me in the future! Thanks again.

[Ezaddeen20]

Why in the world there are 12 dislikes????

maybe they're some jealous math teachers??

God bless you sir.

[fashionista5able]

thank you very much Mr. Patrick! God Bless You, and give you a peace and health.

[JourneyLG]

I don't understand why the limit equals 1, written in red.

[jasonjkeller79]

@JourneyLG Rewrite it as lim n->∞ (n/(n+1))^(1/2) then by dividing the numerator & denominator by n you get lim n->∞ (1/(1+1/n))^(1/2) which goes to (1/(1+0))^(1/2)=1 as n->∞.

[EatShanklish]

Why isnt the first endpoint considered a P-series? Why just the second one?

[jasonjkeller79]

@EatShanklish Because it's an alternating series, thus the need for the alternating series test.

[EatShanklish]

@jasonjkeller79 What I mean was, why is the endpoint a P-series and the second one isnt? They're both 1 over square root.

[jasonjkeller79]

@EatShanklish But they're not both just ∑1/(n)^(1/2), evaluating the original series ∑x^n/(n)^(1/2) at x=-1 is the alternating series ∑(-1)^n/(n)^(1/2) which is not a p-series and requires the alternating series test. Evaluating the original series ∑x^n/(n)^(1/2) at x=1 is ∑1^n/(n)^(1/2) which is essentially just ∑1/(n)^(1/2) since 1 to any power is always just 1, this of course is a p-series with p=1/2.

[TheNexxtGreatThing]

Thanks!!!!! You saved me in my calculus class.

[mpsmonkey]

Great Videos, helped me prepare for finals. :D

[jerstangg]

You can use the alternating series test to test for divergence? I thought it was only conclusive for convergence? I might be mistaken.

[TrakkStar800m]

Nice! I might just pass my test in a few hours! lol

[dcarney999]

So Im not sure why it converges. because 1/ n^(1/2) is less then 1 and we know that if p is greater than 1 it converges so shouldn't that diverges. I am soo confused.

[REVolverTron]

@dcarney999 the LIMIT goes to 0. Contrary to the SUMMATION, then the p-series applies.

[jdizelful]

12 ppl failed calc 2

[mathsman123]

hey patrick

whered u get the 'geeks shall inherit the shirt' from. ia lways wnated something like that. its cool. :) :)

btw u teach maths like how 2Pac rapped.

wait tat doens tmake sense

anways:

Respect respect for the maths professor.

[zubZero505]

Everyone is struggling to understand these concepts before finals. Including me! Once again Patrick comes to the rescue. Thanks bud

[Josh719719]

I have been up since 3:30 a.m, my test calc two mid-term is in four hours, and patrickJMT is saving my ass, once again. love you dude

[kx32ta1]

@Josh719719 You and me both! haha Its 3:35am an I will be up watching his videos until my final at 12:30 pm. Obviously I planned ahead ha.

[crazyesy]

awesomeness! english is my native tongue but im doing my bachelor classes in germany!! this soooo clarified everything, now i can do my assignment! Thanks a mil

[lilbluebuggaboo]

You are Math God!!!

[heidigebhardt73]

i really like your videos

[notslahify]

These videos are the most helpful out of any of the other videos I've been looking at...

[guitarhamster102]

omg dude I LOVE you soooo much. Thank you. You just made my day.

[4yipchoclover]

THANX Patrick! so.. if the limit part DOES NOT equal to 1, do we set the interval of convergence, for example, |x-5|*some# and make that < 1?

[crazyblobie]

you are awesome

[Stephhyc]

Thank you! This was so helpful!!

[jadi929]

wait, in the 2nd example, why did you add 5 to both sides in the inequality again?

[warrioryoyoyo]

You are AWESOME!

[loojiji]

Patrick, you are straight up sexy. I have only one question: How in the world did you become so good at every single kind of math? Did you teach yourself or did you major in it or something?

[MegaBass41]

Excellent video: straight forward and to the point.

If feel like if i see a problem like the one you did I won't have any doubt

on how to do it.

Thanks for sharing these videos.



[xCod3Blu3x]

Just want to say you are a true math genius. Greatly Appreciate the videos. I could honestly skip all of my lectures and still pass the tests much easier than learning from my professor!

[nosoupforyou99]

go sergio!

[vishal075]

Not only do i learn calc but i learnt that sergio sanchez has no problem being the man of faith and a man of science.

[GubitAvitarus]

Thank you very much. I thought I would never understand this, but now in ten minutes I feel confident in the material!

[patrickJMT]

@GubitAvitarus wonderful - that is exactly what i want to happen

[ilion12345]

12 dislikes = 12 profs haha, jk

[SixDayWonder]

I just went through ALL of your videos of Sequences and Series in one weekend to prepare for my test... Gracias Patricio!

[patrickJMT]

@SixDayWonder hope it helps! there are random videos NOT in the playlist, so there is extra stuff if you need it!

[dApimpmx23]

I just want to take this moment and say that your are The Man.

[hockeystick98]

u said that 1/n^(1/2) was convergent, but then 2 steps later u said u was divergent cuz its a p-series. i no it has to be great than one to converge, but because u said 1/n^(1/2) =0, doesnt it mess up the problem?

[gforcegoeswild]

@hockeystick98 when he takes the limit as n goes to infinity of 1/sqrt.(n) its an alternating series and to test if an alternating series converges it has to be decreasing and lim must approach 0

[thirsokewl]

You deserve a medal

[bwhite1726]

It is simply amazing how i can sleep in class and make an A on the test just by watching your videos on the materials we cover in class. Thanks for the videos

[TheBalisto]

at 6:22 if you listen for a few seconds you'll hear a firework going off :P one of the screeching ones...

[billytotheyang]

Wow. Beautifully done and so easy to follow. This is what I need!

[walltowall5]

is there any difference between n=0 and n=1?

or does the method work for both?

[kx32ta1]

There are 106,502 people that are happy you exist.

[salehjamsaljames]

I LOVE YOU MAN, YOU JUST TAUGHT ME AN ENTIRE LESSON IN 10 MINUTES!!! <3

sorry for caps! SOOO HAPPY =D I love power series haha ...

[BramanNoodles]

you, sir, are the batman of calculus.

I'm not entirely sure what that means, but it's a compliment.

[aasolima]

you are AMAZING! (no homo:P)

[amaraspecht]

Our entire calculus class watched at least five of your videos today. All of a sudden it all made so much more sense than how my teacher, my textbook, and my princeton AP Calc study book tried to explain it. What's my teacher getting paid for?? lol GREAT videos.

[patrickJMT]

@amaraspecht sweet! glad i could help your whole class : )

[remirap]

@amaraspecht aside from b/c pat's vids r great, could it also be b/c u reviewed the material

1. teacher

2nd. txtbook

3rd. ap calc study book

3times b4 watching this vid?

[fafajardo09]

isn't it that when it failed the alternating series test, it doesn't necessarily mean that it's divergent?

[aasolima]

@fafajardo09 no, when ut fails the alternating test it means it diverges

[andreasox23]

In the first example does it matter that when he tests the end point and plugs in -1 that the series is only conditionally convergent?? Do you have to say that it is conditionally convergent?

[Glock3d]

@andreasox23 i dont' think u have to say conditionally convergence unless it asks you specifically.

i might be wrong though.

[andreasox23]

@Glock3d

thanks. I'll have to ask my teacher for sure because we have a test on this stuff on tuesday! I don't want to get something so simple wrong, haha.

[Timngu159]

can we get this guy some sort of award or something? SO HELPFUL. i mean, my calc bc teacher is actually really good, she just doesn't have a stop and rewind button!

[areckelh]

such nice handwriting.

[PizzinaldPize]

Excellent video!

[bigkahuna258]

Brilliant! Thank you!

[BreadPS]

This is fantastic I would have never guessed that YouTube had such great math tutorials. I'm so glad my friend recommended it for me. Your video was very clear and easy to understand. Thanks so much!

[daaznnooby32]

u are fricken awesome helped me so much

[macrocec]

straight up, who can dislike this, the guy is superb!!!

[guitarmast3r]

@macrocec there's 11 english majors that watched this video

[xasnkid]

if the lim involving n approaches something other than 1, i assume it just multiplies the x?

[Mulligator]

Does this also work to find the interval of convergence for a power series involving a complex variable z, instead of x?

[o00beSt00o]

MAAAAAAN I LOVVVVVVVVVVVVE YOU FOR THAT !

SIMPLE ! CLEAR ! IT IS JUST AWESOME !

YOU SAVED MY LIIIIIFE :)

[Intensification]

At 7:50, when finding the radius of convergence for x=4, isn't the series convergent because it's an alternating harmonic series? It is NOT absolutely convergent, and that's the only thing you took into factor. You didn't even put the absolutely value when you took away the (-1)^n

[arisiesser]

at 3:12, wouldn't that series be divergent since p=1/2?

[TsuDohNimh2e]

@arisiesser Yes, at that value of X the series diverges, so that is why x=1 is not included in teh interval of convergence

[arisiesser]

at 3:12, wouldn't that series be divergent since p=1/2?

[cantbelieveitsBUDR]

I wish I found out about your videos sooner. I would of had an A in my class instead of my B. Now i know where to look when i take calc 3. Thank you for all your hard work!

[asolorio91]

Thank you so much for your help!! greatly appreciate it! I am now much more confident to go into my final tomorrow morning. You definitely teach loads better than my teacher!

[n3r0t0x1n]

brilliant explanation ... I have seen several of your videos, thank you so much for your good explanations ... awesome !!! Southern USA accent ?

[patrickJMT]

@n3r0t0x1n kentucky boy that does not have shoes or teeth

[gat3way1]

nice handwriting 

[goathead901]

You are a huge help considering I have my Calculus 2 final tomorrow. I just wish I knew about you before now!

[denokindo]

Thanks Patrick. You make my life a lot more tolerable with this series of problems..

[nosaj91321]

honestly i wish all math teachers were this good

[ktothejun]

Tk u very kamsa

[imfinago]

I have my last cal 2 test of the semester at 11 tomorrow. I have been watching your videos on powers series and Taylor/Mclaurin series for the past couple hours and then working examples like yours. Seriously, though, thanks so much for your videos. They really do help.

[j1015379]

Those damn sigma's haha good work literally have a test in 4 hours been crammin!

[FaiththeHairstylist]

funny how i've barely used absolute value inequalities since algebra

[calebbak]

thank you so much. extremely helpful. I have a Russian prof and an Asian TA neither of which i can make any sense of their train of thought but this simplified what didnt make sense for the longest time. letsgoooo midterm

[VileWolfProductions]

Question: When you plug in the -1 to check for convergence, and it turns into an alternating series, doesn't it diverge rather than converge? Because the requirement for convergence for the alternating series of an = 0 is not met, since the limit as n goes to infinity of 1/square root of n does not exist, since p<1, so it diverges, no?

[pvtgarry]

@VileWolfProductions no cause you take the absolute value of it. it's the alternating series test