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.
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.
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)
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.
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.
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
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
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
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."
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?
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.
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.
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.
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.
was it worth it
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.
Yea, the series wasn't expanded correctly but I think most people probably noticed the error so no big deal
+Jason Owen Yes, it was noticed, and it is not a big deal.
Yes it is quite conspicuous
Thanks! I was really confused at first
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)
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.
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.
Same.
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
Technically you need to check x=4 and x= -4 for convergence since the ratio test yields no conclusion when L=1
Agree
Exactly
You plug those points in for x and solve that particular series for convergence.
Yep that's right. In this case both endpoints converge by the p-series test
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
Thanks man, you're a great teacher!
I wish all professors could explain things as simply as you
when n=3, isn't the second term of the denominator 3^4?
it is.
Thanks Ryan Maccombs!! Good video as always :)
you my friend are a life saver
Thank you so much.
but you didn't check the endpoints
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]
@@epicgamermoments4900 Yeah, it doesn't affect the radius of convergence, but he has to check that for interval of convergence
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?
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
I can't imagine life without Khan Academy
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."
Yes
You're awesome sir Sal thanks a lot
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?
Thanks a lot Khan academy
Thank you for doing what you do!
thank u very very very very much
BOTH of what you boxed were the interval of convergence. The radius of convergence is the number 4 itself. Half the interval length
1:40 the third terms is wrong (silly mistake)
I'm in foundations and we defined R = 1/lim sup |c|^1/k
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
but sir what about at point x =4 and x=-4 ...we have to check at end points.....
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.
Yeah, all you do is plug it back in and use some tests to see if it converges or not
Shoutout to Mr. Achille’s class!
0:59 That’s wrong. It’s 4^3 * 3^4.
Thank you! Great work!
thank you so much man, seriously.
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.
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.
Thank you is all I wish to say...
will you please tell me the software name which you are using?
Brilliant!
The fact that he missed n=3 in the denominator in the very beginning makes absolutely no difference here. Stop nitpicking.
Thank you so much
sir,which software is this?
you are king
I never listen to lectures anymore... and my grades couldn't be better.... is that bad lol?!
Great ! 😊
Thank you!
Thanks !
N=3 is wrong
Thanks a lot. . .
W explanation
thank you
Beautiful
thankyou
At the start when you say x=1,2,3.. u mean n=1,2,3...
HOW CAN I KNOW WHICH TEST I SHOULD APPLY WHEN I SEE THE QUESTION I AM GOING CRAZY AHHHH
thank you!!
just exactly what i need lmao
Yay finally get it
what if its, the ones that gives you more than one number?
The radius is 2, because it is always half of the interval
Awesome
wrong im pretty sure, or he made an error in the middle and fixed it later somehow
Oh ho ;)
Let's say...
gonna burn down that useless text book
Yeah, my textbook sucks...
n=1, n=2, n=4? Someone missed a 3
Fix you voice
Thank you