these two infinite series are tricky!

[@subhradeepghosh2523]

I was struggling with a similar problem, this video helped me a lot. Thank you.

[@zelda12346]

LCT is just a more formal version of the Squeeze Theorem, no? I was just thinking along the lines of: For all x in (0,1], sin x < x. since each term of 1/n^2 is an element of (0,1], then sin(1/n^2)

[@zelda12346]

@VeryEvilPettingZoo No, I just confused the Squeeze Theorem with direct comparison because I always remember the direction of implication in direct comparison by thinking about 'squeezing' a convergent/divergent series term for term between it and the constant function above/below it.

[@acrazypunkrocker]

It is also well known that |sin(x)|

[@regulus2033]

At 5:40 you could just use limit (sin(a)/a), while a->0 instead of using Lopital's rule :-)

[@Aviationlover-belugaxl]

I have come up with a similar test for divergence for definite integrals with a singularity at some value. You take the limit as some variable, a, goes to 0 of the area between n and n+a where f(n+a) diverges. That area is like a thin, tall rectangle so its the limit of a*f(n+a). If its zero, it may or may not converge, otherwise it diverges

[@seanfraser3125]

“Which do you like better?” Both are very useful for different situations. So I don’t have a preference.

[@SmileyMPV]

"Very useful" xDxDxD

[@willyou2199]

cos(x) = 1 - x^2 ... immediately obviously divergent. (cos(1/n^2)-1) definitely converges tho, same with sin(1/n^2). Because (cos(x)-1) and sin(x) has form x^k only, (1/n^2)^k only makes each term smaller.

[@zsigmondtelek1612]

For the first sum one one could say that on the open interval (0;pi/2) 0

[@Gathezito]

My test about series is today :p a video like this is refreshing

[@Nick-hc2cs]

hey professor, i recently calculated the derivative of arctanh(sin(x)) and i realized it was sec(x). Does this mean that arctanh(sinx) is the same as ln(secx+tanx)? I plugged it into a graphing calculator and they were the same, except that arctanh(sinx) has extra "lines". Why does this happen?

[@blackpenredpen]

Yes, kzbin.info/www/bejne/iqHPZ6N5rKmqgas

[@Nick-hc2cs]

very cool!

[@EduardoHerrera-fr6bd]

Yo, arctan(x)=ln(x+sqrt(1+x^2). (It's the first thing came out in my head) Sorry, blackpenredpen, I haven't seen that video. Yet.

[@EduardoHerrera-fr6bd]

Damn, I got confused.

[@Kicsa]

Couldn't have made it any easier, thank you professor!

[@rawfries4518]

Have you tried using double sided markers to use up to four colors?

[@neilgerace355]

8:29 writes something while saying something else ... That's clever, I've never been able to do that :)

[@blackpenredpen]

: )

[@TengkuAlam]

i think it would be that little bit easier if we just let 1/n^2 to be u , as n goes to infinity , u goes to 0 and by the limit sin(0)/0 =1

[@specialrel6184]

pls do a video on what it converges to

[@danielnieto7714]

Your videos are amazing. Congrats from spain :)

[@hdckighfkvhvgmk]

But what is the actual sum if it does converge?

[@MrAleguerreiro]

1.48352 according to wolfram alpha

[@eduardomamede6548]

Chen lu!

[@blackpenredpen]

Of course!

[@helloworld5522]

When

[@laylanandao9673]

I'm kinda confused with Chen Lu? Is it Chain Rule? I'm sorry :((

[@kikialeaki1850]

Laylan Dulosa Andao it is chain rule, he was mocking himself.

[@alicwz5515]

5:27 Just do x=1/n^2 Then you get: lim(x->0) sinx/x = cosx = cos(1/n^2)

[@asup759]

don't think that'll always work. pretty sure he only changed n to x since we are more used to differentiating wrt x, and you can't set it equal.

[@0120-c2k]

Why? lim a -> 0 sin(a)/a = 1. (a=n^-2) That's much quicker, and there is no need for LH.

[@GusPintarKalianGoblok]

Pardon me Professor, may i ask a question ? Which is : One the left side equation, when you subtitute 1/(n^2) = x , x should be approaching to 0 while n^2 is going to infinity. I think you should write that way. That's all professor. But your whole answer completely right. Correct me if i am wrong. Hope you succes Professor ☺☺☺👍👍👍Thank You

[@blackpenredpen]

a f I put 1/x^2 I just change x with n. Because d/dn doesn't make much sense since n is meant to be whole number.

[@GusPintarKalianGoblok]

@@blackpenredpen oohh i see. Thanks for explanation Professor.

[@PackSciences]

Could also have presented sum from n=1 to infinity of tan(1/n^2) since it's on the same kind of exercises. It also converges.

[@blackpenredpen]

Yes. I am saving that for my exam problems. : )

[@PackSciences]

@@blackpenredpen Oops, sorry for spoiling your exam problems

[@hokkien_kia]

Cool. What I thought was that for x≥1, sin(1/x²) is strictly decreasing and also bounded as such: 0

[@zafnas5222]

Unfortunately its not so easy, all you have proved is that the sequence sin(1/x^2) converges but we cannot draw a conclusion about the sum. As a counterexample to your logic, we can say the for x≥1, 1/x is strictly decreasing and also bounded below 1/x>0. However, we know that the sum of 1/x from 1 to inf actually diverges.

[@lostboy579]

Best math videos on youtube for calc bc no questions 👍🏽

[@lagrangiankid378]

One year ago you solved the integral of sqrt(tan(x)). Could you solve the integral of tan(sqrt(x))????

[@icespirit]

What if I had infinity product for cos 1/n²?

[@chessi96]

where can I find the microphone you are using? May you do some video about the Lebesgue-integral or measure theorie as well?

[@joaquinbrignoni5769]

Muy bueno! greetings from Uruguay

[@ironmc7972]

I watch this video before I go to school.

[@2neutrino]

Okay

[@pdy9614]

Why does changing it to the x-world matter? isn't that just a personal preference? You can treat "n" as the independent variable and do d/dn instead of d/dx. Right?

[@adityaagarwal3985]

Why cannot we draw any conclusion if lim n-> infinity sin(1/n^2) is 0?

[@avshisharabani1240]

The LCT test is for positive series, how do you know that sin(1/n^2) is positive to every natural number?

[@waynechen8912]

assuming n is positive and greater than one. the value 1/n^2 will always be positive and less than 1. sine of a number positive and less than 1 is always positive.

[@pilotomeuepiculiares3017]

Doens't both series have the same value? I mean, aren't they the same thing but backward? I tried to integrate and it seems like they have the same value

[@a1irice]

Can't you just say that as n approaches infinity lim(sin(1/n^2)) ~ lim(1/n^2) or does that only work if both the numerator and the denominator are infinitesimals?

[@icezhang6169]

Your video is Good, I like your video!!

[@SmileyMPV]

The first is between 1 and pi^2/6 and the second diverges :p

[@SmileyMPV]

Just use sin(x)0 :)

[@codahighland]

@@SmileyMPV That's how I did it, too.

[@sethwest6811]

Im gonna keep it a hunnid i fw the LCT more than the LFD. chen lu!

[@rodrigoaceves9020]

did you just say Chen Lu

[@blackpenredpen]

I certainly did!

[@gilber78]

Now that you’ve shown Σsin(n^-2) is convergent, care to do a video on evaluating it?

[@_DD_15]

How about using Taylor expansion?it's the same

[@5000jaap]

i was watching a video and i got curious, there is any irrationals numbers that sum equals an rational number ?

[@Reliquancy]

pi and (1-pi), but one of a+b or a-b will be irrational if a and b both are might be more what youre thinking of

[@theoleblanc9761]

Use l Hospital rule for calculate the limite that defined sin'(0).... Strange

[@cristianvique4380]

1/(π^n)-(n^π) converge????

[@backyard282]

5:20 Can't we evaluate this limit more simply by letting t = 1/n^2. Now t goes to zero as n goes to infinity, so equivalently we have the limit as t goes to zero of sin(t) / t = 1

[@SeriousApache]

1≠ ☺

[@bheemeshbommireddy4807]

How do I take the limit of sin(1/x) as x approaches 0?

[@thatcallout8223]

This limit doesnt exist. If you put x=0 in the sin(1/x) you will have sin(1/0+) which is sin(+inf) and this have no solution cause the sine function is always between -1 and +1

[@The_Mitchell]

But if cos(x^-2) is < x^-2 for all x greater than ≈ 0.739085 and x^-2 converges from 1 to inf shouldnt cos(x^-2) do the same? Edit: nvm

[@stephenbeck7222]

Your inequality is flipped. cos(x^-2) is greater than x^-2 for all x > ~1.25.

[@The_Mitchell]

@@stephenbeck7222 thanks for some reason i was thinking of cosx2 and x2 lol

[@shreyadixit9999]

Hello sir your voice is very cute and teach very good

[@tgx3529]

sin(x)

[@Patapom3]

Amazing!

[@patrickpayao]

those happy and sad faces ❤

[@VSS031]

Not Chen Lu😂😂😂

[@i_am_anxious02]

1:25 well duh

[@user-xd2qk7yp9k]

I prefer BOTH!!!

[@leonardschreiter6871]

8:40 midlife crises

[@santiagonaranjogallego4592]

I don´t even understand you tshirt :,v

[@EduardoHerrera-fr6bd]

Why is 7 afraid of 8? Because 7, 8, 9. Guy, I'm suppoused to be doing my homework. I saw this eating a mango. Have you eat it?

[@rafciopranks3570]

(x-1)^x=x^(x-1) Can anybody solve that?

[@Mecolo]

Leibniz is the best one

[@sosom-cz1co]

👍👍👍👍

[@Datmexican]

There is 1 dislike Who disliked this???

[@helloworld5522]

Someone hates math, I guess