Radius of convergence using Ratio Test

[@minecraftion25565]

I could've gotten this information in 135 minutes-worth of lessons, but I got it in 7 minutes on Khan Academy (in a way that I can actually understand). Man, I love Khan Academy. It's the only reason I'm passing my calculus class.

[@AD-wg8ik]

was it worth it

[@mrslovvet]

Not that it really matters, but in the yellow line at the top where you expanded the series, the third term has a 4 in the denominator when it should have a 3 (n = 3). Otherwise, awesome video! These really help me study for my midterm.

[@jasonvegan5761]

Yea, the series wasn't expanded correctly but I think most people probably noticed the error so no big deal

[@raycooper1943]

+Jason Owen Yes, it was noticed, and it is not a big deal.

[@sudiptoghosh5740]

Yes it is quite conspicuous

[@frettchen1498]

Thanks! I was really confused at first

[@sathvikswaminathan7933]

the values x=4 and x=-4 also need to be checked since at those values, the ratio test in inconclusive but by the p series and alternating series test, it can be proved that at these two values, the series converges. Therefore x belongs to [-4,4] and not (-4,4)

[@masonbanas9864]

Khan Academy, I can't thank you enough for making something so hard to understand in class so easy to understand from you. What you do is beyond appreciated, please never stop doing what you do my guy.

[@manuelhernandaz2147]

I just want to say thank you for doing these comprehensive videos. My calculous teacher just can’t seem to explain it so it clicks, your really helping me save my grade and learn. Thank you

[@thedarkgenious7967]

thanks. My calc 2 professor went on for 4 hours of drawn out rambling over 2 lectures trying to teach us what the radius of convergence/interval of convergence was and how to find them. I learned more in this 7 minute video than those lectures.

[@chesterfieldtales1340]

Same.

[@magicguy1988]

Technically you need to check x=4 and x= -4 for convergence since the ratio test yields no conclusion when L=1

[@yongbowang1444]

Agree

[@paulyu7739]

Exactly

[@nolanison6642]

You plug those points in for x and solve that particular series for convergence.

[@Grassmpl]

Yep that's right. In this case both endpoints converge by the p-series test

[@cortisol_induced_coma]

I wish all professors could explain things as simply as you

[@krazywyatt2638]

Thanks man, you're a great teacher!

[@zsolttildy5742]

tryna do the question and the same time as sal and lowkey beat him then he starts pulling out the copy paste function and you know its game over smh sal

[@sandeepchoudhary4065]

you my friend are a life saver Thank you so much.

[@kazlee6909]

Thanks Ryan Maccombs!! Good video as always :)

[@Copybook]

I can't imagine life without Khan Academy

[@shreeramaashokaadakoli6784]

You're awesome sir Sal thanks a lot

[@mazahiranjum175]

Thanks a lot Khan academy

[@edrinabenes9754]

when n=3, isn't the second term of the denominator 3^4?

[@windowslogo3577]

it is.

[@Grassmpl]

BOTH of what you boxed were the interval of convergence. The radius of convergence is the number 4 itself. Half the interval length

[@luciazazel2683]

Thank you for doing what you do!

[@hurrdurr25]

but you didn't check the endpoints

[@epicgamermoments4900]

As far as I'm aware, the endpoints don't affect the radius of convergence. The radius is the same whether defined as (a-R, a+R) or [a-R, a+R]

[@priyanshmehta9988]

@@epicgamermoments4900 Yeah, it doesn't affect the radius of convergence, but he has to check that for interval of convergence

[@stumbling]

Is it accurate to state?: "For arithmetic series, the *difference* between consecutive terms is constant. For geometric series, the *ratio* between consecutive terms is constant."

[@monicaelizabeth4069]

Yes

[@mahimakalani2647]

Thank you is all I wish to say...

[@aiicwyt6768]

Shoutout to Mr. Achille’s class!

[@ansarsultan]

thank u very very very very much

[@bowenjudd1028]

I'm in foundations and we defined R = 1/lim sup |c|^1/k

[@terrienichols204]

Why do we not test the end points for convergence? In other words, could it be the case that the interval is not (-4, 4) but [-4, 4] or [-4, 4) or (-4, 4]. Shouldn't we address the endpoints?

[@ZephyrCheez]

You made a mistake in the beginning, the third one should have 4^3 * 3^4, since n is equal to 3 in that scenario

[@cristianferretty6971]

thank you so much man, seriously.

[@cookiesizYumaah]

Thank you! Great work!

[@jackzheng8535]

you are king

[@mechwise6811]

The radius is 2, because it is always half of the interval

[@Vathanaization]

Thank you so much

[@echessbee]

Brilliant!

[@freyappari]

W explanation

[@whoisbobby3]

N=3 is wrong

[@alexwhetstone4464]

thankyou

[@michaelfjmusic]

Extremely helpful video, but I don't understand why he was able to simplify (n + 1)^4 so easily? It just seems too good to be true and I don't fully understand it.

[@annie_adler]

Michael Judd actually, you don't need to expand it in full. As the video shows, the only thing you need to know about (n-1)^4 is that its term with the highest order is n^4. That's enough for you to simplify the limits.

[@ivideos7348]

Thanks !

[@Redblade15]

thank you

[@real_john_doe]

The fact that he missed n=3 in the denominator in the very beginning makes absolutely no difference here. Stop nitpicking.

[@zhangdingkang781]

Thank you!

[@babingurung255]

Beautiful

[@musahamza5829]

Thanks a lot. . .

[@froh_do4431]

how we are going to know that something has to be < 1 for convergence at 6:00? Is that a more complicated statement to show at this moment?

[@tylerdee7529]

The Ratio test says that as long as the limit to infinity of the ratio is less than 1, it converges. After simplifying the limit of the ratio, we are left with an x-term. So basically the function converges if this x-term is less than 1

[@RCstrava]

Yay finally get it

[@jacklazzaro9820]

0:59 That’s wrong. It’s 4^3 * 3^4.

[@MrBoriskaful]

HOW CAN I KNOW WHICH TEST I SHOULD APPLY WHEN I SEE THE QUESTION I AM GOING CRAZY AHHHH

[@nishasingh-yu6ln]

but sir what about at point x =4 and x=-4 ...we have to check at end points.....

[@harrisonbellis8178]

I haven't had lectures on this yet but intuitively I guess you should put x=4 into the expression for the general term of the summation to get a P-series which in this case gives 4/n^4 or 4 times the sum of 1/n^4 which is a convergent P series.

[@candykiss024]

Yeah, all you do is plug it back in and use some tests to see if it converges or not

[@HowViolet]

I never listen to lectures anymore... and my grades couldn't be better.... is that bad lol?!

[@drmichaelsunsschoolformath]

At the start when you say x=1,2,3.. u mean n=1,2,3...

[@curtpiazza1688]

Great ! 😊

[@titusadams5288]

thank you!!

[@arealhumanbeann]

just exactly what i need lmao

[@plasma8321]

what if its, the ones that gives you more than one number?

[@priyanshu715]

1:40 the third terms is wrong (silly mistake)

[@BROBOTwithhope]

wrong im pretty sure, or he made an error in the middle and fixed it later somehow

[@1Naif]

Awesome

[@m.sripadh4338]

will you please tell me the software name which you are using?

[@jinglemich4941]

Let's say...

[@nihalgupta5009]

sir,which software is this?

[@stefanosadane977]

gonna burn down that useless text book

[@rintumondal5658]

Oh ho ;)

[@thenozar7603]

n=1, n=2, n=4? Someone missed a 3

[@marklvrd]

Yeah, my textbook sucks...

[@markwilders5436]

Fix you voice

[@nosirrahonline]

Thank you