Real Analysis 8 | Example Calculation

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:21 Recalling the monotone convergence criterion
00:55 Introducing the example
01:21 Proving monotonicity
04:49 Proving bounded from above
08:55 End result of the example
09:46 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[PunmasterSTP]

Example calculation? More like "Exemplary conversation!" Thanks for presenting the material in such an easy-to-understand and interesting way. I definitely plan to watch this series through to its conclusion!

[cptiglo]

This just looks like my stochastics course
Rly interesting. I dont have much time for youtube but when I am here I rly enjoy ur video

[tatawhillman]

Hello at 8:52 when you used the inequality <= 3. I think that should be a strict < inequality because it’s only equal when n is infinity. Analysis show the highest value of the sequence is e<3 hence my point

[wqltr]

8:20, interesting, my analysis book bounds 1/(k!) under a geometric series instead of a telescoping sum. But this method is also cool.

[none_of_your_business]

Shouldn't the ratio test for monotonicity of a sequence be a strict inequality for <? (2:14) if the ratio of f(n+1)/f(n) where = 1 then obviously it would not be decreasing,
therefore it cannot be true that a(n+1)/a(n) <= 1 means it is monotonically decreasing.

Edit: In the case of equality, it seems the definition is that it is both monotonically non increasing and non decreasing. But would that be the same as monotonically increasing and monotonically decreasing? ;D

A bit of a tricky question, it implies that monotonic increasing and monotonic non decreasing are the same thing, which I don't think that's necessarily the case..

There should exist the term "monotonically constant"...

[synaestheziac]

I tried proving Bernoulli’s inequality by induction and got stuck. I’ve got
(1+x)^(n+1)
= (1+x)(1+x)^n
= (1+x)^n + x(1+x)^n
>/ 1+nx + x(1+x)^n
= 1 + (n+1)x - x + x(1+x)^n
= 1 + (n+1)x + x(-1 + (1+x)^n)

But I don’t know what to do with the big last term. I guess I probably need to use the binomial theorem?

[ichkaodko]

I don't get the first part of proof. In order to determine monotonicity, why we use the an+1/1n and <1 or >1 here?

[dassama]

Hii.. Sir, content not visible due to subtitles. Please keep content above the subtitles.

[中文中国-tu]

when will be update on liner algebra english version please ??? if i purchase your subscr.. can i have access to liner algebra eng version??

[OscarGarcia-ybvl]

min 4:50 why or what is the use of the binomial coefficient?

[kristiyansmilyanski]

For me it's really hard to understand 😔

[gdash]

I think you made a mistake. (n-1)/n is strictly less than 1