Intro Real Analysis 27: Pointwise & Uniform Convergence of Sequences & Series of Functions Examples

Introduction to Real Analysis, Lecture 27.

(0:00) Plan for the coming weeks.
(2:05) Demonstrate the idea of pointwise convergence of a sequence of functions with Mathematica animations.
(2:48) Example 1 (over a few different intervals).
(12:56) Example 2.
(15:03) Example 3.
(21:32) Example 4.
(25:48) Example 5.
(29:12) Example 6. Discussion of pointwise convergence as not preserving nice properties, in general.
(31:20) Define what a sequence of functions is (a function that maps the natural numbers into some function space).
(35:46) Define pointwise convergence of a sequence of functions over some interval.
(37:53) Define uniform convergence of a sequence of functions over some interval.
(42:48) Look at examples again to see when the convergence is uniform or not.
(48:40) Series of functions and pointwise and uniform convergence of series of functions.
(51:47) Example based on geometric series.

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.

[caio]

Great, great, great explanation and didactic skills. Even better than Stephens Abbott's textbook, which I use and always come here for clearer intuitive motivations and understanding of the subject! Really, I'm using abbotts', rudin's, and Ross's and none of those textbooks gave the REAL explanation behind the concepts of pointwise and uniform convergence (and what a seq of function *really* is) like you have done. Would you mind answering why advanced mathematical textbooks do not have this approach? (but at the same time, it is true on my opinion that the majority of the students would learn more and quicker with books in the same way you are teaching). If textbooks are for students and students learn quicker and better with classes like yours, why are math textbooks not like this? I see introductory Linear Algebra books do this (like Gilbert Strang's) but no other advanced math books do. It would make our lives so much easier and better. I study for 3 days (12 hours) about Sequences and Series of Functions, but in only 20 minutes of your videos, I was able to grasp the real concepts. Why is that? Why there's never visual and geometric explanations on those textbooks if *those tools help so much*? Thank you!

[algorithmo]

is there a video on fourier series convergence

[manauwarhussain]

could you tell me this is which university ???

[samario_torres]

can you provide the link for source code

[kimberlyafonso]

hi can u tell me which text book do u use??