Convergence and Divergence: The Return of Sequences and Series

We learned a little bit about sequences and series earlier in the mathematics course, but now its time to work with these some more, now that we understand calculus! First up, what does it mean for a sequence or series to be convergent or divergent, and how can we tell which one it is? Let's find out!
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[@emmaell6659]

Your voice is so clear and stress-relievingly good to hear.

[@saeedahmed-9290]

I had a problem all during my semester in this concept.. but now it's so clear !
I can't thank u more man :)

[@calculus8399]

​@@jonathanlimjun6238 what do you know about convergence

[@harismasoom6634]

PROFESSOR you explained it so beautifully.
Thanks.

[@debajyotisarkar1145]

Thanks professor Daves.
This video really helped me in understanding the basics of convergent and divergent series.
And , I would also like to place a demand for a video on Maclaurain's series.

[@ProfessorDaveExplains]

I did that too, check the mathematics playlist!

[@samitkumar1170]

Just the type of explanation I was looking for. Perfectly explained.

[@mathew_pang]

at 8:50 why that sum will be divergent ? i thought it will converge to 1/5?

[@IslamicLifestylevibess]

same here

[@Vinickw]

This is the limit of the sequence, not the series. It means that for a very big _n,_ the sequence will be approximately {...1/5, 1/5, 1/5...}, and adding infinitely many 1/5 will result in infinity.

[@Sarah-re7cg]

Omg, thank you so much. This video really helped bridge the gap between all this information and how it works together. Before, it was like a jumbled mess in my head.

[@Sarah-re7cg]

@@jonathanlimjun6238 yeah, professor Dave is really great at explaining topics in a much more coherent and intuitive way

[@Kiky_MedPhysicist]

Thank you sir for your dedication and for making this free! 🙏

[@AnAzsproduction]

Thank you for giving that example! I would love to see some more as you continue to do these videos

[@ClumpypooCP]

"let's converge a little"

[@juraijbinjamal5313]

Good one

[@f3ralp1g3on6]

converge me daddy UwU

[@The-Cookie]

@@f3ralp1g3on6 ayo- 🤨🤚

[@riyabiswas906]

Thankyou sir for such a simplified explanation.

[@davideveringham3827]

(6:35): for r=-1, the terms are a,-a, a,-a, a,-a… .

[@GodbornNoven]

Then series diverge when they sum constants and ever increasing values and converge when they sum ever decreasing values? really simple and good explanation!

[@jawwadshahed]

Very good explanation. Thank you so much for all your nice demonstration.

[@annaV513]

Thank you for this video, it has lifted up some uncertainties.

[@AlessandroZir]

thanks, man!! really clear and useful, and also insightful; ❤️❤️

[@iyasuwalalign2860]

how could i explain my love to this wonderful man

[@hailabseyoum5746]

great lecture! Thank you professor a lot!

[@angadthandi2281]

@@jonathanlimjun6238 Nice bot

[@supreethmv]

1:16
This seems like a divergent series but actually is convergent. There are some simple and complex proofs to show that 1+2+3+.....= -1/12.
This is same as Riemann zeta function for -1.

[@chaosend3815]

Great video! your explanation is very easy to understand

[@nelayapu5789]

thank you very much sir....this concept is clear to me now

[@niekbeijloos8355]

Very clear explanation, thanks!

[@MathCuriousity]

Would someone mind explaining why it is that every power series is a Taylor series? (I know every Taylor series is a power series but I’m wondering why the reverse is true - and without invoking heavy analysis stuff)!

[@nnofficial2414]

It was a good revision! Great job 👏

[@geektoys370]

This man can explain a 3 month university subject in 10
Minutes

[@vianekabanda8348]

This professor is good. I like this video

[@douglasposho9299]

Thanks for the knowledge you have shared with me.

[@curtpiazza1688]

Wow ....its all coming back to me!!

[@sandeepanpoddercse3819]

thanks sir

[@nononnomonohjghdgdshrsrhsjgd]

i like everything in the way how you explain!

[@MissOk777]

Great explanation !

[@salaa23]

Pirates if they give up Piracy and go to the Caribbean University 6:05

[@TanJunYu]

thank you so much!

[@ECEAbhishekKumar]

Thank you 😊👍 so much from India.

[@zeinebkefeya1946]

Perfectly explainedd🔥

[@sidharthmohanty6434]

Thanks loved it.

[@danielgilder8672]

This helped me with my Alevel

[@reengaesh7545]

Thanks so much 🙏🏼🙏🏼🙏🏼I have understood a lot 😊

[@jontedeakin1986]

whats the name of the theorem at 7:17?

[@niaala]

Thank you so much.really helpful

[@irfankadem3628]

GREAT WORK KEEP UP!

[@nweniaung9621]

Thank.You Professor

[@beetaff7841]

Sir , limit of1/n is zero still series is divergent this theorem goes wrong in this case is it exception ??????? Pls reply sir

[@MimiMimi-ss7jk]

Thaaaaanks a lot🙏Finally,I understood!

[@ckarcher4504]

7:05 how to get that answer?

[@mahinduliyanage1583]

Pro Dave u are the best........................!!!!!!!!!!!!!

[@harrisonbennett7122]

Thank you

[@fernanmontevista204]

Hi prof Dave .do you have videos about Taylor and Mclaurin series.Thank you.

[@ProfessorDaveExplains]

Yep!

[@мир-н3л4х]

which playlist is this video in?????

[@ProfessorDaveExplains]

math

[@leejunzhao]

2:54 the sequences should start from 1

[@fishlasagna7368]

Thank you 🙏🙏🙏

[@shahrukhkhanshahrukh3504]

great man

[@umararshad9567]

convergent = limit does not exist

[@ccbgaming6994]

First equation = -1/12 ;)

[@thestar001Official]

I lost it at
"In fact,..."
7:10
😪

[@jah_141]

this stuff is confusing idk if i can do it man but i have to for school

[@sedeqalsakkf9315]

I confused how lim of 1/2^n = 1 . I thought it should be 0. becz 1/inf = 0.

[@RickyRicheRebelle]

he just said in the video clearly that if a sequence converges to 0 as it approaches its infinite-th term, the series that the sequence generates would have a finite number limite. i think u're mixing up sequence and series, which is the first thing he stated in the video

[@ericard4165]

Hey professor Dave, what else after calculus in this course

[@ProfessorDaveExplains]

linear algebra! and then hopefully differential equations.

[@ericard4165]

thanks they're really helpful

[@JillianRiccio-rv3nu]

🙏🏻 🙏🏻 🙏🏻

[@imbreakingdown2683]

2:17 Why Am I laughing so hard

[@joongi9839]

I needed help on precalc 20 and it started talking about integrals and limits 😥😥

[@ProfessorDaveExplains]

i have a tutorial earlier in my mathematics playlist on sequences and series, that's probably the one you're looking for. just go to the long playlist, it's somewhere in the 80s i think.

[@wjrasmussen666]

If you are going to reference previous tutorials, a link in the comments would be good.

[@ProfessorDaveExplains]

just go to my mathematics playlist, also i usually link using cards in the top right

[@murphy6177]

来自华东地区某211高校，看完这个视频才知道国内高数老师是什么东西😅

[@thisisnyx]

This Math Is Worse Than Programming.... My Head Is About To Explode

[@sanjaycosmos9679]

4:37 .1+2+3+....=-1/12 ...ramanujuan....

[@SurajSharma-pv3di]

It is a special case for a special type of function. You will get zero no. If you write this in your exam

[@scptime1188]

Only using analytic continuation on a different function (continued riemman zeta function) that is closely related to but not interchangeable with that sum. The function that gives you that sum (riemman zeta function) only works for inputs with a real part greatrr than one. The RZ function breaks when you get smaller than one. But if you "continue" the riemman zeta function throughout the complex plane (ensuring it is analytically continued), when you plug in (-1), which using the non-continued riemman zeta function (∑«n=1→∞» (1/n^s) for varying values of s) would blow up to infinity, you get (-1/12). So there is a sense in which the sum you get from the non continued RZ function, 1+2+3+4... also equals (-1/12).

[@PratimMukherjee]

Simple af

[@adventuresofgaara8936]

I'm here from FLVS (Florida Virtual School) and I would like to say Big Chungus Funny.

[@gaminisiriwardana5946]

Sir still I haven't got a reply. Waiting for your explanation. Creating a youtube clip for those questions will be realy helpful. Waiting for your email

[@sehreenrani9608]

best

[@dareto2181]

수학이랑 영어를 합하니 배로 좆같네;;

[@adriaaan.n]

Jujutsu kaisen brought me here

[@Mr_Passionate]

[@codexsreyans9141]

Its wrong, as the series n^2/(5n^2+4) would be a convergent series.......please don't try to teach till you are not fully clear in your concept

[@ProfessorDaveExplains]

It's a convergent sequence, but it does not converge to zero, so the series that is the sum of the terms of the corresponding sequence will be divergent. Please don't try to make presumptuous comments until you are fully clear on the concepts.

[@johnoconnor2648]

You should first understand the relation between a SEQUENCE and a SERIES before making presumptuous comments...

[@darknightcorexd7227]

this video has false information,a limit of a sequence can exist and be equal to a real number and the sequence can diverge.

[@ProfessorDaveExplains]

if a sequence converges at some finite number, it's convergent. that's what "converge" means. a divergent sequence goes to infinity, by definition.

[@mainulislam6765]

Professor Dave I just came across your video and I want to express my appreciation at the very clear presentation of the concepts; not only are the slides very clear, you speak with a certain clarity which is rare to come by in a mathematics tutorial. Thank you .

[@samjames1253]

Thank you so much!!!

[@jamvibes3550]

Hi prof dave, your channel helped me a lot in my study. I humbly for a request a video explaining the fourier series. I will wait for it. Thank you so much.

[@32_gurjotsingh82]

refer to 3b1b video on fouries series...

[@ericard4165]

Like your animations

[@pluto4301]

0:50 bold of you to assume that 😅

[@sachinrajpandey5242]

When the limit of a sequence is zero then the corresponding series will be convergent and when the limit of a sequence gives some constant value L then the sequence is convergent. Is this right? Please correct me.

[@Ashish-hl7rk]

so if the resulting series is zero it is convergent?

[@courageandpeace1944]

love from India

[@TheBrammiej]

Hey Dave. Would you consider following up this course with a linear algebra course? All the best, Bram

[@ProfessorDaveExplains]

don't worry, linear algebra is coming! i already filmed some of it.

[@TheBrammiej]

Professor Dave Explains Hugs!

[@duelingo8731]

you reduce my stress levels by 10000%

[@rinr3300]

I don't get it because at the beginning he said that if it gets to a finite number it is convergent and in the end when he gets1/5 he said it is divergent ;-; please somebody help me

[@Attalla-t9b]

In the beginning, he was talking about sequence. At the end, he was talking about series (It seems like they have different criteria for divergence/convergence)

[@Bhanuday-pp1dj]

Love from india

[@nabanitagoswami6652]

You are very helpful 😄😄❤️....

[@shankarabhat7505]

Hey can a convergent sequence have an answer which is a complex number...as n approaches infinity

[@ProfessorDaveExplains]

hmm yes i believe so, at least definitely if there are complex numbers in the sequence!

[@TheBluePhoenix008]

you gotta do more of these man

[@badreddineadrham9605]

1+2+3+4+5+……= -1/12
😢

[@noahgsolomon]

He knows a lot about all kinds of stuff

[@Diviner-ip2pq]

❤❤❤❤❤❤❤❤❤❤❤

[@hammadullahshaikh4735]

❤

[@koamivikeledome]

🙏

[@antonioochoa2673]

you're a goat for this video bro, respect g

[@johnholme783]

Crystal clear! Thank you!

[@Has_Chemist2.0]

How can math be amazing like this 😍

[@julytdalida4860]

Is it possible for a sequence with two different numbers to converge?

[@JoanKemon]

THANK YOU SO MUCH

[@إيمانإيمان-ف4ي]

Excellent explanation