in times of this pandemic, when the prof doesn't care much to buy a stylus pen and write down the scratchwook, and just reads out from his 1000 years old pdfs... I am left scratching my head, as to how did the first line of the proof statement came to be! Thanks for being a great help and teacher!!
Bro, you explained everything in a crystal clear manner. It's people like you that make the lives of students easy. Keep up the good work. And lot of thanks!
I have my Analysis 2 final tomorrow morning. This video really helped me. I've never thought about finding the max of the function for this kind of proof (I always struggle to find an n not depending on x), that's one more trick that I can use. Thank you!
dear kind sir you explained it so well and so clear, you helped me tremendously. Your channel is a true gem for undergrad level math (hard to find these days)
thnx :D helped alot!! was not explained (just vaguely ) in the book (all the rest in the book is so good explained though), so now I understand how to do it!! :D
The operators in math usually suffice as an indication of a matrix product. Whole the generality comes from the sufficient statistics associated with equality over non continuous nob mutually exclusive sets.
It usually takes a crash course I. Euler and archimedes with polynomial techniques to work out most legibus of math. Problem solving olympiads usually about formattibg designs I. Matlab tbf
You can use the Arithmetic-Geometric inequality applied to the denominator instead of the first derivative test. en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means
This video is well done and makes a lot of sense! Right now I'm working with the sequence g_n (x) = 1/n tan^(-1) (n^2 x). Deriving gives n/(1 + n^4 x^2). I don't know how to solve for x to maximize this function, what can I do instead? Thank you!
That was pretty useful for me thanks. However we should me more careful, since x is not necessarily positive, I think that it's not enough finding M such that f_n(x)
that video was great, thank you. I have one question left. How do we know the local maximum you calculated is the global maximum? Couldn't there be any points lower than - sqrt(1/n), which would have values greater than our max value?
Exactly right, derivative provides only local info. The valid approach would argue that lim as x approaches \pm \infty of f_n(x) goes to zero showing that indeed the local maxima found are also global.
We can use the fact that 1+nx^2≥2 times absolutvalue(x) times sqrt(n). And the fact that Absolutevalue(x/(1+nx^2)) ≤ absolute value of(x)/(2 times sqrt(n) times absolute value(x)) For x not equal to zero. We will be done.
@waterbearsandwich Sure. We have |x/(1 + nx^2)| = 1/|nx + 1/x|. Note that x and 1/x have the same sign, so 1/|nx + 1/x| = 1/(n|x| + 1/|x|). Now as |x| >= 0 we can use AM-GM to get denominator >= sqrt(n). Hence, 1/denominator
Is that not a local maximum? How do you know it is a global one? Would the function then not have to be bounded? What precisely is the use of setting it less than or equal to a local maximum?
Actually, you are not looking for the maximum of the function. You are looking for the absolute maximum of the function. That is given by the maximum metric
Because it's a sequence I would draw a graph first against the sequence. If the sequence is an increasing one then f_n-1(x) - f_n(x) = differentiable manifold. And erase for n and differentiate against dn. Rembering to equate to epilson so long as axiomatically epilson is less than 0 - would imply an increasing n.
in times of this pandemic, when the prof doesn't care much to buy a stylus pen and write down the scratchwook, and just reads out from his 1000 years old pdfs... I am left scratching my head, as to how did the first line of the proof statement came to be!
Thanks for being a great help and teacher!!
Bro, you explained everything in a crystal clear manner. It's people like you that make the lives of students easy. Keep up the good work. And lot of thanks!
thanks man!
I have my Analysis 2 final tomorrow morning. This video really helped me. I've never thought about finding the max of the function for this kind of proof (I always struggle to find an n not depending on x), that's one more trick that I can use. Thank you!
You are very welcome, good luck!!
All of your video are greats! I'm physicist and you helped me a lot to understand all these maths in simple words!thank you!
You are welcome!
dear kind sir you explained it so well and so clear, you helped me tremendously. Your channel is a true gem for undergrad level math (hard to find these days)
thank you!!
This was so helpful. I now understand the definitions and theorems.
Excellent!
Watched the whole video and it helped a lot. Thank you so much, I love your videos. =)
Thank you!
Thank you sooooooooo much for your video
You saved my life
thnx :D helped alot!! was not explained (just vaguely ) in the book (all the rest in the book is so good explained though), so now I understand how to do it!! :D
great!!
very helpful and easy to understand
Thank you!
Life saver! Thank
You
You always will have the math video I want!Thanks
Great explanation, thank you
That was very much useful, but the problem you chose is kinda "cheated". Try this one for example: n*x^2/(n+x)
:)
Ya very True
The operators in math usually suffice as an indication of a matrix product. Whole the generality comes from the sufficient statistics associated with equality over non continuous nob mutually exclusive sets.
Hey keep up the great work
What an amazing video 👍
thanks!!!
thanks man!!!
It usually takes a crash course I. Euler and archimedes with polynomial techniques to work out most legibus of math. Problem solving olympiads usually about formattibg designs I. Matlab tbf
Keep up the great videos friend.
Thanks man:)
This is fantastic! Thank you so much. I think I can do my homework now! :)
You can use the Arithmetic-Geometric inequality applied to the denominator instead of the first derivative test. en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means
Great explanation thank you!
This video is well done and makes a lot of sense! Right now I'm working with the sequence g_n (x) = 1/n tan^(-1) (n^2 x). Deriving gives n/(1 + n^4 x^2). I don't know how to solve for x to maximize this function, what can I do instead? Thank you!
That was pretty useful for me thanks. However we should me more careful, since x is not necessarily positive, I think that it's not enough finding M such that f_n(x)
Beautiful!! thanks a lot!
that video was great, thank you. I have one question left. How do we know the local maximum you calculated is the global maximum? Couldn't there be any points lower than - sqrt(1/n), which would have values greater than our max value?
Exactly right, derivative provides only local info. The valid approach would argue that lim as x approaches \pm \infty of f_n(x) goes to zero showing that indeed the local maxima found are also global.
very nice, thanks
You are welcome!
We can use the fact that
1+nx^2≥2 times absolutvalue(x) times sqrt(n). And the fact that
Absolutevalue(x/(1+nx^2)) ≤ absolute value of(x)/(2 times sqrt(n) times absolute value(x))
For x not equal to zero. We will be done.
Thanks for you
You are welcome
Thank you !
Which software do you use for this board ?
smoothdraw
By the way, you could've just used AM-GM on the denominator to bound f_n.
cool
@waterbearsandwich Sure. We have |x/(1 + nx^2)| = 1/|nx + 1/x|. Note that x and 1/x have the same sign, so 1/|nx + 1/x| = 1/(n|x| + 1/|x|). Now as |x| >= 0 we can use AM-GM to get denominator >= sqrt(n). Hence, 1/denominator
Is that not a local maximum? How do you know it is a global one? Would the function then not have to be bounded? What precisely is the use of setting it less than or equal to a local maximum?
Thank you
How can I study the Uniform Convergence for the series of function ∑(x/(x^2+1))^k
Where x is from R
Isnt that "max" point for fn(x) just local? What about the value for very negative x-values?
They have a derivative that is negative, and never zero. It becomes arbitrarily close as you approach the infinities, but they aren't in R.
Thank u
You are welcome
So finding max is the same as finding the lim sup?
Do you have an example using the Tn test?
How do you know to use 0 in absolute value difference? Shouldn't it be justified, how do you do that?
Wow. Just wow.
thanks man!
Most people got the idea that they should reason like a Turing machine I can't explain why I don't.
Actually, you are not looking for the maximum of the function. You are looking for the absolute maximum of the function. That is given by the maximum metric
Hello Sir, plzz proof uniform convergence for this ques. By defination fn(x)=x^n(1-x) where
x€[0,1]
best
Thank you😃
apply Mn test
I thought there is a difference between f_n(x)
A limit.
I say it's bounded uniformly.
If f_n(x) is increasing or decreasing the it can't. First principles would use epilson or Kronecker delta
It's a correction issues
Because it's a sequence I would draw a graph first against the sequence. If the sequence is an increasing one then f_n-1(x) - f_n(x) = differentiable manifold. And erase for n and differentiate against dn. Rembering to equate to epilson so long as axiomatically epilson is less than 0 - would imply an increasing n.