Limit Comparison Test
This calculus 2 video tutorial provides a basic introduction into the limit comparison test. It explains how to determine if two series will either both converge or diverge by taking the limit of the ratio of the two sequences to see if it equals a positive finite number. Examples and practice problems include series that can be simplified and compared to a p-series or a geometric series.
Integral Test For Divergence:
• Calculus 2 - Integral ...
Remainder Estimate - Integral Test:
• Remainder Estimate For...
P-Series:
• P-series
Direct Comparison Test:
• Direct Comparison Test...
Limit Comparison Test:
• Limit Comparison Test
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Alternating Series Test:
• Alternating Series Test
Alternate Series Estimation Theorem:
• Alternate Series Estim...
Absolute & Conditional Convergence:
• Absolute Convergence, ...
The Ratio Test:
• Ratio Test
The Root Test:
• Root Test
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Series Tests - Practice Problems:
• Calculus 2 - Geometric...
Taylor & Maclaurin Polynomials:
• Taylor Polynomials & M...
Taylor's Remainder Theorem:
• Taylor's Remainder The...
Power Series - Interval Convergence:
• Power Series - Finding...
Power Series - Derivatives & Integrals:
• Power Series - Differe...
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Think you made a mistake in the third one, isn't the limit as n approaches infinity of 1/3^n equal to zero ? Please confirm.
They were talking about sum of 1/3/^n@@onwabadosini2962
@@onwabadosini2962 indeed the limit as n approaches infinity of 1/3^n =0 but in the 3 problem we are limit comparing (1/3^n +5 * 3^n /1) = lim(3^n / 3^n + 5) and that equals 1
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I recommend watching the direct test first and then the limit one because it’s easier to understand
100% true
u are 100% correct
I crushed calculus 2, thanks to this channel, and now I'm in graduate school enjoying these as a quick reviews of the tedious mechanics of calculus
Whenever I have confusion, I search your video and understand what going on. Thanks bro!
0:22-1:22 For anyone who is confused about why the last question is divergent when the limit=1, this explains it. This is an important rule to remember, it is simply a proof. The limit doesn't determine the answer. It just proves that the answer found for one series (bn in this case) is going to be the same for the other (an).
Thank you!!
thanks! couldn't have said it better
You mean the one before last?
You may actually be the reason i pass my engineering maths test this coming Monday. Really helpful explanation video. I had not yet come across such clear and concise explanation. Thank you
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Awesome video!!! I love the way you teach us like you have a God-given gift. God bless you you are a grade saver! I literally don’t even care that I don’t get the material in the class cuz In the end you are my real teacher always! ✍🏻❤️
You're the OG homie. Thank you once AGAIN for enlightening us peasants about another math concept. Keep rocking man!
Thanks! I was looking for how to solve one of these problems algebraically and what you did made perfect sense.
Professor Organic Chemistry Tutor, thank you for another great video/lecture on the Limit Comparison Test in Calculus Two. This is a very important test in determining convergence and divergence of a series. Learning different tests for convergence/divergence can be a challenge for all students in Calculus. Thís is an error free video/lecture on KZhead TV with the Organic Chemistry Tutor.
I have a test on this Monday and I plan to binge watch these videos and make a much needed A on this test!! Thank you sooo sooo much!!
fail
Last minute studying isn't gonna work. You should've studied earlier. I should take my own advice too. My test us tomorrow. So yay
Organic chemistry is a life saver
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Dude thanks again. I was struggling with this, but thanks to you I’m back on track. 🐐
I love you, never gonna stop loving you, youre getting me through calculus and into an engineering degree.
Thanks you so much for making these videos! They really help!
Amazing video, literally running through my math problems because of your extraordinary explanations! Thank you!
how do we know if we should use the limit comparison test or the direct comparison test?
your voice is so soothing .i'm so happy i found your channel
Thank you again for the 56th time for helping me understand and review my calculus, physics and organic chemistry. If I pass my semester I owe it to you.
Did you pass the semester
@@Daniilope I did 😁 I almost didn't pass organic chemistry but I made it. This channel helped me pass Calculus 1, Calculus 2 and physics.
@@arianeparadis6439 that’s so good to hear! Currently calculus is kicking my butt so hopefully this channel works it’s magic like it did on you 😭🙏
@@Daniilope Don't give up! Calculus can be difficult to understand. If you understand the reason behind using a certain formula or type of limit you'll find it easier to remember. It's a lot of practice. Wish you the best!
Thanks for the video! You are an awesome teacher!
last problem can be solved using direct comparison test really appreciate your effort man, thanks.
Your videos are immensely helpful and have gotten be through Organic Chemistry and now Calculus II. I would have failed without you so thank you!
What did you make in orgo? i take it next semester
You are the best professor I have ever seen . ❣
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Thanks again OCT. I can now sleep soundly in my bed knowing how to execute the LCT. You are the best.
When you plot the series, both converge to the cero. 1/n as n is greater and greater the sum is approaching to cero (0)
You are awesome, sir. Keep it up!
In the second example, you have to calculate the first term of the series, n=0, therefore obtaining the series that starts at n=1. After that, you just have do exactly what JG did about the harmonic series. When you see that the new series is divergent and the limit test gives you 1, you can unify the first term to the series that starts at n=1, and then obtain the one the you had a the beginning.
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Thank u sooo much sir for ur kindness and for helping us
I have my calculus 2 final in 1 hour. Sir, you just cleared my doubts about the limit comparison test. Thank you for so much.
I'm curious too...can't just leave us hanging dude
I guess you didn't learn the limit comparison test as well as you had hoped.
How did you do?!?
Well, I graduated college 2 months ago but I did pass the test. I remembered that I got an 87 on that final and now I work in tech. Success story 😂
@@kelvincheng999 you give me hope, I have a calc 2 midterm tomorrow and hopefully will pass.
taking 4 summer classes. have my second calc exam next Wednesday. if I pass this exam, actually no. if I pass this semester in calc 2, I owe it all to you. calc 2 in 10 weeks is a big beast but im currently kicking its ass
Thank you so much for this
Thank you for this video! Could you please do another video like this but working with e?? Most of my homework and practice quizzes have those types problems.
you sir have blessed me
Thanks my guy, I usually watch khan, but this filled a lot of gaps in my understanding that i couldn't get with khan.
what if you are using the limit comparison test for a series that isn't a rational expression (for examples, ln(sec(1/n)), how would we go about this problem?
Got my final in 6 minutes and ofc this guy saves the day again haha
excelente video, muy practico para el aprendimiento rapido.
just a heads up, you have the harmonic series at 5:49 going from 0 to inf.. but its undefined at zero
Can anybody suggest where I can see more such examples as shown in this video using this specific technique ?
You are so helpful!
2:33 we now finally know what Gunna and Future were talking about when they said they were pushing P, they were talking about Calc 2 Limit Comparasion test, p=3. They're ahead of the rap game so much man, Drake's out there sampling Alvin and the Chimpunks and they're out here sampling Limit Comparasion Tests and Differential Calculus. Kudos to them and OCT! 🤲🤲
Excelent explanation sir
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wouldn't the example at 8:17 be solvable through the direct comparison test?
So would the first example.
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Why do you multiply the numerator and denominator with 1/n at 6:55?
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Thank you, but I think the summation in the series b_n should starts from 1 not from zero.
You are my math god
Can anyone please explain the step at 6:55? When he multiplied the top and bottom by 1/n. Thanks!
if you understand that, can you please explain it to me?
@@bilinmezevren-291hzu He's basically multiplying it by one but in a different form. 1/n divided by 1/n is 1. The denominator of sqrt(1/n^2) actually simplifies to 1/n. The reason that he puts it under the square root though, is so he can multiply the radicands and just keep it under the square root.
U help me a lot thanks for that
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On example 2 I’m confused how you chose to divide by 1/n and sqrt of 1/n^2
@The Organic Chemistry Tutor how would u remove significance if the numerator has a trig such as sin(n)
let's say u had summation from 0 to infinity (1+sin(n))/10^n. how would dis be solved using removal of insignificant stuff.
for the second example, why are you able to use 1/n when if sub n=0 , it is not finite which violates one of the constraints which says that the expression must be positive an finite for all values for n
You are a life saver, Thank you !
يسطا لو فهمتها ياريت تشرحهالي عليا اختبار بعد بكرا ده الانستا بتاعي . (_5stc) ولو ملقتهوش خد ده endl9
@@moazosama معلش والله مخدتش بالي من الريبلاي إلا دلوقتي سوري😂. أتمنى تكون أديت كويس.
The Organic Chemistry Tutor you are too much
so are you saying that if you get a finite number for the limit comparison then both are the same as the results you got from the direct comparison"?
for those who didnt understand the part in #2 where he removed the 5 in the limit, you can use L'Hopital on the limit which is ln3*3^n/ln3*3^n which cancels out to 1
thank you!
What is the some real-life application of comparison test?with examples.
so, for the limit comparison test you do not need to compare if the new series b_n is bigger or smaller than a_n?
Thank you sooooo much
Thank you 🙏🏻
Bravo, maestro
Aww! I literally just had my test 3 hours ago! It’s okay tho, I still love you ♥️
Hope you scored high!
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Sir how to take bn in limit comparison test
Isn't the limit comparison test flawed for some examples? If we were to have the sum from n=1 to infinity of 1/(n^2-1) and tried to evaluate whether it converges or diverges, the closest thing to compare it with would be the sum from n=1 to infinity of 1/(n^2). Using the limit comparison test, we would see that they also reach some value L, so that would mean that since (1/(n^2)) is convergent, 1/(n^2-1) should also be convergent. However, the first value of the sum in 1/(n^2-1) is 1/0, which should make it impossible for them both to converge. Are there some rules that I have missed?
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For the first example, since is bn is a bigger series and converges couldn't you say the an series converges by regular comparison test?
4:47 not to mention 1/n^3 is bigger than a of n so DCT would work too
8:16 (Wait for bn diverges, may I ask how does that make a sub n diverge? I know we have a finite number in the end). But isn't it a rule that if bn converge then a sub n converge. and if a sub n diverge, then b sub n also diverge? Doesn't it not work the other way around for both conditions. So how is it that b sub n diverge = a sub n diverge? Also thank you for your videos, they really help get things into perspective for calculus! :D
I don't understand the second example, where the first term is already divergent. Shouldn't the series start with n=1? In this case there was no problem but when we use 1/n^2 for comparison, (1/sqrt(n^4+1) on the left)we will get different answers(convergent if starting from n=1, divergent if the series starts with n=0) which is probably not the case in the series we want to test. What is wrong with this situation?
You are amazing!!!
man thankyou!!
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n can not equal zero as your bn in the second example, in order to correct this i believe you should set both starting at n=1 and then add one to the n in the an.
couldnt you use the direct comparison test for the last example? Because 1/(3^n + 5) is less than 1/3^n and 1/3^n converges, so by the direct comparison test, 1/(3^n + 5) would converge?
bingo
Thanks sir
Thank you, sir.
FAIRA Fa Hey, @8:53, do you know how 1/3^n became (1/3)^n? Thanks
@@uzumakikuroko9000 yes because 1^2=1, 1^(2000)= 1 1 exponent anything is one. So you still have the same term 1/(3)^n I hope it helped.
FAIRA Fa yeah, that’s true. Thank you for your help, I was just kinda skeptical whether or not it is “legal” to go from 1/3^n to (1/3)^n. But your explanation made it clear, so thanks again☺️
how do u know whether to use direct comparison or limit comparison? is it just what you see first or preference or something
It’s more of a preference, you can use whatever test you find easier
Thank you
love you bro
does an have to be smaller than bn?
If you watched the direct comparison test video he has, hes subtly nudging that test when he goes to prove b_n or an as either convergent or divergent.
can anyone explain 4:39 STEP where we have limit as 1?
I'm just wondering, but isn't the last problem technically a Direct Comparison Test?
@@mundheral-ahmadi8939 I was thinking the same thing. I think he's just taking it one step further.
thank you