Power Series & Intervals of Convergence
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- Опубликовано: 14 окт 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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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.
SUCH a clear explanation and excellent examples! What a great teacher!!
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 😂
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
Wow, i was struggling to grasp this in class but this made it so clear. Thank you
Earned a sub, your Calc II content is the best I've seen on this platform.
Thank you Dr. Bazett. Just learnt/understood what these terms are..beautifully explained.. 🌻
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!
I always liked how you put advanced math concepts in perspective easily and smoothly. Connecting the dots greatly enhances my understanding. Thank you!!!
Haven't seen better explanation than this! thanks
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 love the video, I couldn’t understand this Power Series and watching this video made amazing clear.
Glad it helped!
I can't beiieve you explained this so well in nine minutes. Thank you!
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 ?
Just a clear explanation. Thank You, sir!
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?
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.
Thanksss a lottttt sir....You're explanation is better than my prof...
Looking for a great explanation for this topic.Good thing i found this one .Thank you very much doc.
he makes this topic so easy I almost forgot how much I struggled in class
He loved my comment! I love you too sir!!!
Thank you teacher
YOU ACTUALLY ATE THIS UP THANK YOU SO MUCH WOW
God bless best explanation on yt
Oh so what I gleaned Only now is that a power series is like adding more variables, sort of combining the maths of functions with the maths of series and sequences right?
Great video. Small mistake at 8:40 you say diverges unless x=0 when it should be x=2.
great explanation
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. 🤔
you helped me so much. thanks😀
Becoming your fan day by day sir❤️
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👌
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?
very good explanation keep up the good work
Can we also use the ratio test to find the Radius of convergence? like R=lim of absolute value of an+1/an?
Hi Trevor ..Please do videos on topology and functional analysis
amazinggg explanation. thank youuuu so much
bro you are like jesus christ of calculus you saved me nglllll
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
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.
Good work
What the fffffffffunctiona , love it
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.....
Amazing explanation 👍👏 ....
Hi
I have a question about "Convergence of a Power Series"
Can you help me solve that?
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
I am a JEE apsirant and i really wanted to know something more about interval of convergence
Amazing!
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
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🤣🤣
At 8:39 , x = 2 turned into x = 0, I don't understand
Same☹️
I think it's a small mistake from Dr. Bazzit weldone for spotting it.
I think he meant to say: "it only converges if (x-2) =0" not "It only converges if x=0"
Doctor please correct us if we're wrong
thanks a lot mannn
Legend
8:40 editor got confused or something that's supposed to be "Series diverges unless x=2. SO R=0"
I don’t understand about n/n+1 ?!
200 SAAL JIYO AAP SIR JI
|1-2| =1 why was -1 one plugged in?
Thanks!!!!!!
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.
the graph is vague but ok
Useless