Power Series & Intervals of Convergence
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- เผยแพร่เมื่อ 28 ก.ย. 2024
- Power series are series of the form c_n (x-a)^n where the c_n is a sequence and x is thought of as a variable. Whether it converges or diverges depends on the x. By applying the Ratio Test, we decide the interval upon which the Power Series converges, as well as the "ratio of convergence" and the "center" of the power series. The results turn out very analogous to Geometric Series.
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SUCH a clear explanation and excellent examples! What a great teacher!!
Wow, i was struggling to grasp this in class but this made it so clear. Thank you
I love the video, I couldn’t understand this Power Series and watching this video made amazing clear.
Glad it helped!
Earned a sub, your Calc II content is the best I've seen on this platform.
it's 3 am, I have my exam at 8am and you might've just saved me... Thank you so much!!! Great explanations and clearly shown writings for each step. The work you do to get to each step takes a little bit to understand but not too bad at all!
did u pass ??
@@中文中国-t4u been 3 years bro
@@evrenunal3644still stands
How was your exam?
@@evrenunal3644 we still wanna know 😂
Loving your calc II vids! I have a question though, if we run into an interval where R=infinity and we converge for all x, what about if x= infinity? Would that change the ratio depending on how you would then solve the limit?
YOU ACTUALLY ATE THIS UP THANK YOU SO MUCH WOW
Yo I'm pretty sure you lectured my math 122 class like a month or something ago, if this was actually you I loved your lecture, if this isnt I loved you video regardless👌
Sir at 5:52 you say for x=1 it coverge by A.S.T
But A.S.T is summation [ (-1)^n-1*bn ] so we check bn value for convergence
But in the example summation [(-1)^n /n] not (-1)^n-1 then how are we using it sir
you helped me so much. thanks😀
why does my textbook say that you don't need to take (x-a)^n into account when using ratio test? Wouldn't these also work? It says you just need to do the ratio test of C_n
Amazing explanation 👍👏 ....
I am a JEE apsirant and i really wanted to know something more about interval of convergence
Hi
I have a question about "Convergence of a Power Series"
Can you help me solve that?
amazinggg explanation. thank youuuu so much
At 9:07 , the radius of convergence R can only be 0 or 1 right? As we are using ratio test?
@@DrTrefor oh right, my bad
thanks a lot mannn
the graph is vague but ok
Thanks!!!!!!
200 SAAL JIYO AAP SIR JI
Useless
Videos like these make me cry. Just so thankful how God has gifted you so much and you still care to share your gifts with others. Thank you.
Haven't seen better explanation than this! thanks
At 8:39 , x = 2 turned into x = 0, I don't understand
Same☹️
Your videos are amazing Dr. Bazett. I really love your enthusiasm as you speak, and I believe that enthusiasm permeates into the mind of the viewer as well.
Thank you!!
One of the best math explanations I’ve ever heard
Thank you Dr. Bazett. Just learnt/understood what these terms are..beautifully explained.. 🌻
Thank you teacher
God bless best explanation on yt
I don’t understand about n/n+1 ?!
Just a clear explanation. Thank You, sir!
Amazing video ! I wanted to know whether this is useful for determining the radius of convergence of the Taylor series, for example for ln(1+x) ; should we analyse the remainder function or the radius of convergence or both ?
What if I have the following power series : (x-a)^cn where c is some constant or (x-a)^n^2 how to do these?
Thanksss a lottttt sir....You're explanation is better than my prof...
bro you are like jesus christ of calculus you saved me nglllll
8:40 editor got confused or something that's supposed to be "Series diverges unless x=2. SO R=0"
Great video. Small mistake at 8:40 you say diverges unless x=0 when it should be x=2.
What the fffffffffunctiona , love it
I think for the first example i is [ -1 , -3 ] not [1 , 3]
No it is correct center =2 so [1,3] watch previous video for clearity
@@vijayvarma5501 i had an exam at the I commented and I passed the exam so I don't remember what was this about and I will never watch one of these series videos again ever🤣🤣
Legend
great explanation
I really wish my teacher took the time to explan things like you do here. I assume you already know how much help you are to kids like me in college and I just gotta say I love you Dr.Trefor, please never stop making content like this!
Good work
Are there any restraits on c_n? Does every number in the series have to be a real constant for this to be considered a power series?
While the focus in my calculus courses is indeed that c_n is a real number, the same concept certainly works for other types of coefficients, most prominently the c_n being complex numbers.
he makes this topic so easy I almost forgot how much I struggled in class
He loved my comment! I love you too sir!!!
Looking for a great explanation for this topic.Good thing i found this one .Thank you very much doc.
can you please solve the following:
summition n ranges from 0 to infinity ((X^2+3)/12)^n , using root test .... find interval and radius of convergence.....
My professors script says that when using the ratio test for power series its limit L gives us the radius of convergence as 1/L for all L of (0,+oo). So it would mean that for a limit of 1 the radius of convergence would be 1 not inconclusive as you said. 🤔
Thanks sir for your extraordinary explanation really we are gifted you by the God to understand the topics of math in such a easy way God bless you and keep growing up.
Can we also use the ratio test to find the Radius of convergence? like R=lim of absolute value of an+1/an?
Last example : why do you say that 0 times infinity is 0?
Plugging in x=2
You get the limit of something that is always 0 no matter what the value of n. The limit of 0 is, well, 0
In contrast, something like e^-n × n
is the product of something approaching 0, and something approaching infinity. This is different, and you need to compare their growth rates (using L'hopital's or something), in order to determine the 'winner', so to speak.
very good explanation keep up the good work
I can't beiieve you explained this so well in nine minutes. Thank you!
Hi Trevor ..Please do videos on topology and functional analysis
Becoming your fan day by day sir❤️
|1-2| =1 why was -1 one plugged in?
Amazing!
My college cal II teacher never actually went through explaining how to tell if the divergent end is equal to the parameters or not. You, sir, taught me in six minutes! Thank you so much!
I always liked how you put advanced math concepts in perspective easily and smoothly. Connecting the dots greatly enhances my understanding. Thank you!!!