Society needs more women like this: educated, radiating confidence, and living life 🙌🏽❤️. This was a phenomenal lecture, and really ties gracefully into the previous one on why the radius of convergence makes sense in the first place.
This is beautiful explanation of the Radius of Convergence that is base on the Ratio Test. Professor Breiner your performance is off the mathematical charts.
This video was so helpful. It made this concept so simple. I cannot thank you enough. Also helped with the basic algebra a ton, which can be the hardest part sometime!
Corrections previously made by Jose Orton and Mike H: In the fourth exercise at the minute 15:18 of this video, the power series shown corresponds to cosh(x). (hyperbolic cosine) Verification: The Maclaurin series of e^x is: e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + x^5/(5!) + x^6/(6!) + x^7/(7!) + x^8/(8!) +…… The Maclaurin series of e^( - x ) can be obtained substituting - x in the above series: e^( - x ) = 1 + ( - x) + ( - x)^2/(2!) + ( - x)^3/(3!) + ( - x)^4/(4!) + ( - x)^5/(5!) + ( - x)^6/(6!) + ( -x)^7/(7!) + ( - x )^8/(8!) +…… I got: e^( - x ) = 1 - x + x^2/(2!) - x^3/(3!) + x^4/(4!) - x^5/(5!) + x^6/(6!) - x^7/(7!) + x^8/(8!) +…… And hyperbolic cosine is: Cosh x = (1/2)[e^x + e^( - x )] I substituted the series of e^x and e^( - x ) in the above formula: Cosh x = (1/2)[ 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + x^5/(5!) + x^6/(6!) + x^7/(7!) + x^8/(8!) +…… + 1 - x + x^2/(2!) - x^3/(3!) + x^4/(4!) - x^5/(5!) + x^6/(6!) - x^7/(7!) + x^8/(8!) +……] Cosh x = (1/2)[ 2 + 2x^2/(2!) + 2x^4/(4!) + 2x^6/(6!) + 2x^8/(8!) +…… ] Cosh x = (1/2)2[ 1 + x^2/(2!) + x^4/(4!) + x^6/(6!) + x^8/(8!) +…… ] Cosh x = [ 1 + x^2/(2!) + x^4/(4!) + x^6/(6!) + x^8/(8!) +…… ] Cosh x = Summation from n = 0 to ∞ of [ x ^ (2n) / (2n)! ] My name is Carlos Vicente Dominguez. I am a graduate student of the specialization in electric power systems at Central University of Venezuela in Caracas. Best regards from Venezuela.
+Dominic Dill Think about it like this: Divide both the numerator and the denominator by n. Then, we have 1/(1+1/n). As n approaches infinity, you can see that 1/n approaches zero and the fraction therefore approaches one.
If I ever went to university, MIT's one of the few schools (honestly the only one I have in mind) that I would attend if accepted. Their level of academic instruction is GOLDEN 🏆
This is a recitation, and covers the material in more simplicity for students having trouble. Lectures also go through basics, because that is how a class of Calculus 2 or BC is supposed to be. People who want can always take more advanced classes or accelerated honors classes for which most of MIT probably are a part of. Like any other college MIT, has a large assortment of different majors from Business to International relations to the more stereotyped MIT major Engineering.
OMG THANK YOU, I was trying to determine what to do when lim n-->infinity of some n times x and it finally got explained at the end and i havent found it anywhere!
omg thank you so much, I just needed a simple explanation of how to find the radius of convergence and every other video I came across just finds the interval of convergence, which I understand, but doesn't go over the radius
Thanks for explaining that the limit of (x/n+1) is 0 because x is fixed and outrun by n+1. My textbook just skips directly to 1/n+1, which made me believe I couldn't do basic algebra.
dandaman113 is surely right, it is missing the (-1)^n factor. Other than that, an absolutely explicit and concise explanation of radius of convergence.
I don't think that Ex. #4 is the cosine series because in the Taylor series of the cosine the sign alternates between (+) and (-). For instance the X²/2! term should be negative. That is not the case in Ex. #4. EDIT: I saw below that someone already pointed this out. Ex. #4 turns out it is the series for cosh(x).
You are making me confused. In one statement you are saying that radius of convergence is limit a(n+1)/a(n) and in another statement you are saying 1/ limit of the previous value. 🤔
In multivariable calculus, is the radius of convergence an actual radius? It is odd that it would be called a radius if there was not a situation in which we were in some way talking about a circle.
pls pls help me understand the significance of elliptic integrals.If someone has found some good material pertaining to the topic please redirect me to it.it will be too much helpful to me.I want a complete understanding of the topic. I'll be thankful for the help.
ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-5-exploring-the-infinite/part-b-taylor-series/session-97-power-series/. Best wishes on your studies!
i want to travel usa to give a hug to all the mit teachers, seriously
Society needs more women like this: educated, radiating confidence, and living life 🙌🏽❤️. This was a phenomenal lecture, and really ties gracefully into the previous one on why the radius of convergence makes sense in the first place.
This is beautiful explanation of the Radius of Convergence that is base on the Ratio Test. Professor Breiner your performance is off the mathematical charts.
What a beautiful way of teaching this subject, thank you Christine!
Crystal clear about the concept after watching - you're amazing, Thank you :)
I learned more in 18:01 than I learned in an hour of class!
this helped me. and i actually found mistakes made in lecture notes by my prof. absolute life saver !
This video was so helpful. It made this concept so simple. I cannot thank you enough. Also helped with the basic algebra a ton, which can be the hardest part sometime!
Corrections previously made by Jose Orton and Mike H: In the fourth exercise at the minute 15:18 of this video, the power series shown corresponds to cosh(x). (hyperbolic cosine) Verification:
The Maclaurin series of e^x is:
e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + x^5/(5!) + x^6/(6!) + x^7/(7!) + x^8/(8!) +……
The Maclaurin series of e^( - x ) can be obtained substituting - x in the above series:
e^( - x ) = 1 + ( - x) + ( - x)^2/(2!) + ( - x)^3/(3!) + ( - x)^4/(4!) + ( - x)^5/(5!) + ( - x)^6/(6!) + ( -x)^7/(7!) + ( - x )^8/(8!) +……
I got:
e^( - x ) = 1 - x + x^2/(2!) - x^3/(3!) + x^4/(4!) - x^5/(5!) + x^6/(6!) - x^7/(7!) + x^8/(8!) +……
And hyperbolic cosine is:
Cosh x = (1/2)[e^x + e^( - x )]
I substituted the series of e^x and e^( - x ) in the above formula:
Cosh x = (1/2)[ 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + x^5/(5!) + x^6/(6!) + x^7/(7!) + x^8/(8!) +…… + 1 - x + x^2/(2!) - x^3/(3!) + x^4/(4!) - x^5/(5!) + x^6/(6!) - x^7/(7!) + x^8/(8!) +……]
Cosh x = (1/2)[ 2 + 2x^2/(2!) + 2x^4/(4!) + 2x^6/(6!) + 2x^8/(8!) +…… ]
Cosh x = (1/2)2[ 1 + x^2/(2!) + x^4/(4!) + x^6/(6!) + x^8/(8!) +…… ]
Cosh x = [ 1 + x^2/(2!) + x^4/(4!) + x^6/(6!) + x^8/(8!) +…… ]
Cosh x = Summation from n = 0 to ∞ of [ x ^ (2n) / (2n)! ]
My name is Carlos Vicente Dominguez. I am a graduate student of the specialization in electric power systems at Central University of Venezuela in Caracas. Best regards from Venezuela.
+Dominic Dill
Think about it like this: Divide both the numerator and the denominator by n. Then, we have 1/(1+1/n). As n approaches infinity, you can see that 1/n approaches zero and the fraction therefore approaches one.
Dear sister, thank you for your good performance and I pray to God to help you in the service of humanity
((Egyptian mathematics teacher))
Her enthusiasm is admirable
If I ever went to university, MIT's one of the few schools (honestly the only one I have in mind) that I would attend if accepted. Their level of academic instruction is GOLDEN 🏆
This video really helped clear up the idea of radius of convergence for me thanks!
This is a recitation, and covers the material in more simplicity for students having trouble. Lectures also go through basics, because that is how a class of Calculus 2 or BC is supposed to be. People who want can always take more advanced classes or accelerated honors classes for which most of MIT probably are a part of. Like any other college MIT, has a large assortment of different majors from Business to International relations to the more stereotyped MIT major Engineering.
I could listen 👂 all day 🙂
Amazing Teacher and Professor
OMG THANK YOU, I was trying to determine what to do when lim n-->infinity of some n times x and it finally got explained at the end and i havent found it anywhere!
your explanation is very vivid. I can understand. Thank you...Hopefully one day, I can study in MIT..
omg thank you so much, I just needed a simple explanation of how to find the radius of convergence and every other video I came across just finds the interval of convergence, which I understand, but doesn't go over the radius
The series in the last example is actually hyperbolic cosine, aka cosh(x), which is not alternating like cosine :-)
i don't how to think u ... u really useful and easy to follow
CORRECTION::: EX 4 is the series of cosh (x)
Jose Orton ....yes brother it's coshx
Thanks for explaining that the limit of (x/n+1) is 0 because x is fixed and outrun by n+1. My textbook just skips directly to 1/n+1, which made me believe I couldn't do basic algebra.
I love this teacher! Amazing explanation!
Thank you very much!!!
I am preparing for AP Calculus BC and it helps me a lot!!!
a real math teacher, wish binghamton had those
dandaman113 is surely right, it is missing the (-1)^n factor. Other than that, an absolutely explicit and concise explanation of radius of convergence.
I Really Like The Video From Your Ratio Test -- Radius of Convergence
Amazing woman
Hello I am from India and I really like teaching and mits professors
Good lec and explanation method is also good
Ma'am this video was a great help to me!
Thanks alot for posting MIT lectures :)
Not that it changes the end result, but that last power series is not cosine, but the hyperbolic cosine.
Cosine series alternates in sign. Your series has all terms positive. (Ex 4)
I think it is cosh instead of cos.
Is the hyperbolic cosine, so that is the reason why all signs are positive
Wish my professor could explain it like this.
Thanks from Ga!
My thoughts exactly! Having a professor that can communicate this well is priceless.
The limit of 1/n does not exist because it diverges;however, 1/n^2 converges to 0.
Damn she is a very good teacher I just understand it easily
I don't think that Ex. #4 is the cosine series because in the Taylor series of the cosine the sign alternates between (+) and (-).
For instance the X²/2! term should be negative. That is not the case in Ex. #4.
EDIT: I saw below that someone already pointed this out. Ex. #4 turns out it is the series for cosh(x).
If you are life up to now im say you Thank you teacher ❤️
easy to catch and love it
Mam, Thank you so much for being helpful to us.
this video helps me a lot
really appreciate
Very helpful video. Thank you for streaming this. Really helped me!.... Lovely instructor too :D
@dominicdill take the limits dude: consider lim x--->infinity of x/(x+1) = lim----->infinity of 1/(1+1/x) = 1
Thank You for your honest.
wow you are so helpful! You are much clearer than my professor. Thank You :)
thank u for this video , I am not american person , In addition, I don't speak english well ... but I always follows you , thanks
Shes amazing
Yes, this was informative!
Example four was not cosx as it was not an alternating series.
You are making me confused. In one statement you are saying that radius of convergence is limit a(n+1)/a(n) and in another statement you are saying 1/ limit of the previous value. 🤔
Great explanation...
To divide (x/2)^(n+1) by (x/2)^n she unpacks that expression instead of simply observing that (y^(n+1))/(y^n) = y ... I wonder y? 😀
Amazing 😁🤩
It was very informative. Thank you.
You're awesome
For the third example, why is the limit as n approaches infinity of n/(n+1) = 1? Wouldn't this be infinity over infinity which does not simplify to 1?
The Lim sup of root test is a better way to find the radius of convergence.
Wow, so so SO informative. Thank you so much! :)
(on EX. 4, the series is the function - absolute value of SIN(x
not just SIN(x)
this helped sooooooo much
Great Teacher
Very helpful. Thank you!
9:33 those are monomials
Infinity factorial @11:20...😀
very intelligent woman.
Sure was!
thanks !!
please upload video for the differential equations (second order)
finding the limit of 1/n gives one and that of 1/n^2 is also one. so which of the two converges of diverges?
great HELP
what did she say about the 2nd example? X^n/n! It's the Taylor's series for what? Eliax?
you are awesome :) Thank you!
awesome awesome awesome!!!!!!!!!!!!!!
Ma'am please solve this problem series m=1 to infinity (X)^(log m)
I don't understand the last part? how to pull mod x squared out times the whole series part .Help?
Also insane test's
Can anyone tell me is this is what is taught in MIT in UG courses?
Thank you. Thank you.
Is Christine still teaching? Where can I find her lecture?
In multivariable calculus, is the radius of convergence an actual radius? It is odd that it would be called a radius if there was not a situation in which we were in some way talking about a circle.
I think it becomes a 'radius' when you use complex numbers for X, I'm not sure.
Isn't the last series cosh(x) and not cos(x)?
Lol I had example 2 on my exam yesterday (no I. D.o.n.t go to MIT).
I got it right 😄
Please upload a video about TRACING OF CURVE PLEASE!!!!!!URGENTLY!!!!!!!!!!!!!!!
can you do a problem using trig functions
Luckies. :p We have to test the end points and get the interval of convergence too. Lameeee. Thanks for the great video though .
Yea, I did get a little sloppy in my wording. Thanks
reminds me of leslie winkle from the big bang theory
Example 1 just use the root test...
Thanks you mum... Love you........ That all i need
nice video.........
why (2n)! equals to 1?
they don't?
i would understand that if u drag z camera little down so that i can see ur 00 clearly
Wasn't limsup?
pls pls help me understand the significance of elliptic integrals.If someone has found some good material pertaining to the topic please redirect me to it.it will be too much helpful to me.I want a complete understanding of the topic.
I'll be thankful for the help.
Which chapter of the course is this?
ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-5-exploring-the-infinite/part-b-taylor-series/session-97-power-series/. Best wishes on your studies!
@@mitocw Thanks a ton!
Is this for University students or High school pupils?
This video is for anyone who needs to brush-up on this concept.
***** Ok, very helpfull anyway, thank You!
turns head* sniffs*
I'm golden :)