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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[@emmaell123]

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

[@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 .

[@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.

[@random6033]

the sequence converges to 1/5 so the sum isn't finite
EDIT: wait that comment is old

[@magiic3871]

Second reason I think is cause that rule where if r is less than 1 than it’s convergent doesn’t apply here cause this isn’t a geometric series.

[@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.

[@samitkumar1170]

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

[@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!

[@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

[@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.

[@Kiky_MedPhysicist]

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

[@duelingo8731]

you reduce my stress levels by 10000%

[@AnAzsproduction]

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

[@geektoys370]

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

[@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...

[@annaV513]

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

[@riyabiswas906]

Thankyou sir for such a simplified explanation.

[@AlessandroZir]

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

[@jawwadshahed]

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

[@Has_Chemist2.0]

How can math be amazing like this 😍

[@davideveringham3827]

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

[@ClumpypooCP]

"let's converge a little"

[@juraijbinjamal5313]

Good one

[@f3ralp1g3on6]

converge me daddy UwU

[@The-Cookie]

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[@nnofficial2414]

It was a good revision! Great job 👏

[@katunduchabala7266]

Thanks for the video. This has made me understand a lot of things I have been missing. Good work!

[@antonioochoa2673]

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

[@TheBluePhoenix008]

you gotta do more of these man

[@chaosend3815]

Great video! your explanation is very easy to understand

[@hailabseyoum5746]

great lecture! Thank you professor a lot!

[@angadthandi2281]

@@jonathanlimjun6238 Nice bot

[@iyasuwalalign2860]

how could i explain my love to this wonderful man

[@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!

[@zeinebkefeya1946]

Perfectly explainedd🔥

[@ECEAbhishekKumar]

Thank you 😊👍 so much from India.

[@vianekabanda8348]

This professor is good. I like this video

[@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.

[@niekbeijloos8355]

Very clear explanation, thanks!

[@AhmadShah-oz5wu]

Love from Pakistan, Taxila
Sir ❤❤

[@nelayapu5789]

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

[@johnholme783]

Crystal clear! Thank you!

[@reengaesh7545]

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

[@salaa23]

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

[@MissOk777]

Great explanation !

[@إيمانإيمان-ف4ي]

Excellent explanation

[@pluto4301]

0:50 bold of you to assume that 😅

[@douglasposho9299]

Thanks for the knowledge you have shared with me.

[@nabanitagoswami6652]

You are very helpful 😄😄❤️....

[@nononnomonohjghdgdshrsrhsjgd]

i like everything in the way how you explain!

[@jontedeakin1986]

whats the name of the theorem at 7:17?

[@noahgsolomon]

He knows a lot about all kinds of stuff

[@irfankadem3628]

GREAT WORK KEEP UP!

[@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!

[@curtpiazza1688]

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

[@ericard4165]

Like your animations

[@MimiMimi-ss7jk]

Thaaaaanks a lot🙏Finally,I understood!

[@mahinduliyanage1583]

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

[@samjames1253]

Thank you so much!!!

[@JoanKemon]

THANK YOU SO MUCH

[@sidharthmohanty6434]

Thanks loved it.

[@beetaff7841]

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

[@nweniaung9621]

Thank.You Professor

[@fernanmontevista204]

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

[@ProfessorDaveExplains]

Yep!

[@leejunzhao]

2:54 the sequences should start from 1

[@niaala]

Thank you so much.really helpful

[@ckarcher4504]

7:05 how to get that answer?

[@Oscar-gx2yf]

Thank you Lord Farquad you are the best!

[@fishlasagna7368]

Thank you 🙏🙏🙏

[@courageandpeace1944]

love from India

[@sandeepanpoddercse3819]

thanks sir

[@user-cr8bd9qk1r]

Very very helpful!!!!! 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)!

[@TheIronLife]

This helped me with my Alevel

[@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!

[@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)

[@julytdalida4860]

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

[@shahrukhkhanshahrukh3504]

great man

[@jamebello9869]

so helpful

[@мир-н3л4х]

which playlist is this video in?????

[@ProfessorDaveExplains]

math

[@thestar001Official]

I lost it at
"In fact,..."
7:10
😪

[@Ashish-hl7rk]

so if the resulting series is zero it is convergent?

[@KamalCabdinasir]

Yes

[@harrisonbennett7122]

Thank you

[@robel_asefa]

👏 great

[@thisisnyx]

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

[@Jillybeanz2007]

🙏🏻 🙏🏻 🙏🏻

[@Bhanuday-pp1dj]

Love from india

[@umararshad9567]

convergent = limit does not exist

[@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

[@ccbgaming6994]

First equation = -1/12 ;)

[@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).

[@imbreakingdown2683]

2:17 Why Am I laughing so hard

[@raghavtandon8033]

Ty for making me understand bro

[@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

[@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😅

[@koamivikeledome]

🙏

[@adventuresofgaara8936]

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

[@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

[@hammadullahshaikh4735]

❤

[@cavendis3423]

nice

[@Diviner-ip2pq]

❤❤❤❤❤❤❤❤❤❤❤

[@PratimMukherjee]

Simple af

[@jah_141]

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

[@badreddineadrham9605]

1+2+3+4+5+……= -1/12
😢