What series convergence test do I use?
2024 ж. 14 Мам.
359 848 Рет қаралды
What series convergence test do I use? This video will help you with your calculus 2 class. Best of luck! Get the file and the notes first 👉 / 102348401
More practice: • Which series convergen...
Timestamp: (BIG THANKS TO Treanungkur Mal)
0:00 getting started
4:38 Question 1
9:07 Question 2
14:00 Question 3
23:40 Question 4
30:20 Question 5
38:41 Question 6
44:40 Question 7
49:37 Question 8
53:10 Question 9
59:18 Question 10
1:05:25 Question 11
1:10:04 Question 12
1:16:03 Question 13
1:21:03 Question 14
1:28:43 Question 15
1:30:35 Bonus Question
Get the file and the notes first 👉 www.patreon.com/posts/102348401
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Number 14 Can also be expressed as sum from 2 to inf of 1/(n^2-1)
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Can anyone help me with the timestamps? Thank you.
4:38 Question 1 9:07 Question 2 14:00 Question 3 23:40 Question 4 30:20 Question 5 38:41 Question 6 44:40 Question 7 49:37 Question 8 53:10 Question 9 59:18 Question 10 1:05:25 Question 11 1:10:04 Question 12 1:16:03 Question 13 1:21:03 Question 14 1:28:43 Question 15 1:30:35 Bonus Question Guys the Check pdf in the description for the question. Give a like if you liked the hard work ❤️
@@treanungkurmal803 Thank you so much Treanungkur!
@@blackpenredpen Really pleased sir that you replied ❤️
My professor allows us to bring a notecard to the test and I wrote the series convergence tests table you provided on mine. I'm feeling pretty confident, thanks!
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watching this in 2023 a day before my calc 2 exam. thank you so much for your videos!! they really help me. btw i'm from nazarbayev university, kazakhstan
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I'm currently doing real analysis and this video was soo helpfull thanks so much
44:22 thanks for the shout out my guy
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Your pdf is very very helpful to me sir thank you for your effort for us.
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For number 3, the root test would work as well, and includes a famous limit (n!)^(1/n) / n as n approaches infinity.
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For the ratio test, I always like to organize the equation by term, so you get (-2)^(n+1)/(-2)^n
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I have a test on Sequences on Series due today (I'm taking Calc 2 online through my local community college since I'm still in high school) so I'm using this video as a final review. I already watched it through once a few months ago when I was teaching myself, but I'm back again!
Abbreviation may change between teachers for the final test explanation (problem 4), but this is very helpful! Thank you bprp
Timestamp 4:38 Question 1 9:07 Question 2 14:00 Question 3 23:40 Question 4 30:20 Question 5 38:41 Question 6 44:40 Question 7 49:37 Question 8 53:10 Question 9 59:18 Question 10 1:05:25 Question 11 1:10:04 Question 12 1:16:03 Question 13 1:21:03 Question 14 1:28:43 Question 15 1:30:35 Bonus Question Check the pdf for the questions
Integrate. [Cos^-1x (√1-x^2)]^-1 / Log{1+(sin(2x√1-x^2)/π}
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almost
Sir from where I can download the pdf of those questions ?
In question 4, why did you replace sin(2n) by 1? Thank you!
For bonus question, can use the double angle formula to change it to 2sin²(1/2n). Last we recall, there is a similar question earlier where we use limit comparison test for 1/(4n²)
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Students the _"scary sigma"_ ∑ ☠ takes a lot of practice to master and BPRP has done a marvelous job here putting this material together covering all the various tests you need to know. ✔ *BonusQ:* LCT: [sin(1/2n)]^2 ,which is monotonic↘, with lim n→∞ of p-conv (1/(2n)^2), let ⌀=1/(2n) so lim ⌀→0 [sin(⌀)/⌀]^2=1.
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4) by direct comparison, the sum is between the sum of 1/(n+3^n) and -1/(n+3^n). Both series are convergent by limit comparison test with 1/(3^n), which is convergent by gst Convergent by direct comparison, limit comparison, and geometric series test Limit comparison test actually works in most cases where the limsup and limslow are defined numbers. Of course, most isn’t good enough.
For 12, you can just do direct comparison test with pi/(2x^2) which converges as arctanx is bound by pi/2.
This is good practice for my final!
On the bonus question at the end, a bit of analysis can also be used. You can use the comparison test with the sum of 1/(2*n^2), which means the series converges to a value less than or equal to pi^2/12. The reason for this is because you can use the series expansion of 1 - cos(1/n) to get the sum of 1 - (1 - 1/(2*n^2) + O(1/n^4)), which simplifies to the sum of 1/(2*n^2) - O(1/n^4). It's less than or equal to the sum of 1/(2*n^2).
Even easier: notice that 1-cos(1/n)=2sin²(1/(2n)). This converges by #9 on this video, which showed that sin²(1/n) converges by the limit comparison test with 1/n². Since sin²(1/n) is always positive and it converges, then it does so absolutely. Since it converges absolutely, then any rearrangement of its terms also converges absolutely, and to the same value. Therefore, the sequence of even terms of sin²(1/n), sin²(1/(2n)), must also converge. Or you could just do limit comparison with 1/n² again lol. You basically did the same thing anyway, but it's even easier if you just use the trig identity so you don't have to worry about the Taylor Polynomial approximations and such.
Thank you so much!
Integrate. [Cos^-1x (√1-x^2)]^-1 / Log{1+(sin(2x√1-x^2)/π}
Thank you 🌈👌
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Yo blackpenredpen! For your next 100 problem challenge, could you tackle trigonometric equations and identities and stuff? Basically, formulas like sinC + sinD, the half angle formulas, and all that good stuff used in simplifying and solving problems. Thank you for everything.
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5) limit of this over n^(-3/2) is 0. n^(-3/2) is conv by gst. This is convergent by LCT
13 Question , i think it is conditionally Convergent as |Un|=1/n(3+1/n) where by p series p=1
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Preparing for analysis 1 exam in IT For the number 12 I did a Limit comparison test with 1/n²+1, which converges because it's less than 1/n² which, again, converges because it's a p series where p>1. So from the LCT I got π/2>0, so the 12 converges too. The solving should be correct, so yay I found another way which, to be honest, seems way easier than getting the graph, taking the integral et cetera.
If the ratio test is inconclusive, you might be able to use Raabe's test (of course it's not certain that it'll be conclusive either). You could add it in the notes after the ratio test.
It basically says that if the ratio limit is equal to 1, calculate the lim of n(1-an+1/an),let's say that this limit is equal to L, then if L>1 then the original series converges,if L
Excelente!
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I am studying for the CSET 3. Thanks for your help. Do you have any study guides I may use, please?
Hey, thank you for your videos, just the link of the file says it got blocked by bitly, do you have another one?
Thanks
5 can also be done by integral test
How I can send a PDF to you sir?I am your big fan from India and regularly watch your videos on calculus and other topics .After watching out this video I am able to proof the p series convergence for any p>1. I want to show it to you sir. Love you and your videos so so much...!!!!!
Ratio test is called d’alembert rule in 🇫🇷
For the bonus, I used Taylor series. Cos u = 1-u^2/2. So 1-cos(1/n) goes to 1/2n^2. Thanks to the p-series, because p is equal to 2 in this one, the bonus serie converges.
Cos(1/n) at n tends to inf =1 hence divergent
Bonus Question: 1-cos(1/n)=2sin^2(1/2n) multiply divide by 4n^2 take limit sinx/x=0 as x-->0 we will be left with 1/n^2 variable term which we can take as vn now un/vn=1/2 hence by P series Vn is convergent and so is Un! :)
Can we do the same absolutely convergence test for complex valued sequence?❤
How to limit X tends to zero sin(X) also tends to zero. In problem no 9
Now that I've learnt calculus 2, I can do these questions. Question 12 looks challenging looking at the live chat just guessing and using integral test. I would say use direct comparison test. The series is obviously less than or equal to the series (pi/2)/(n²+1), which is less than or equal to the series (pi/2)/n², which converges by the p series test. By direct comparison test, original series is smaller so it converges as well Ok nvm integral test can be used but in my exam i have to show that it is positive, continuous and decreasing to use it. Don't really need to show it is positive side it is obvious but we have to show that the limit as x goes to a is f(a) to show it is continuous. Then to show it is decreasing we either use a_n
Can you solve for convergence/divergence using the alternating series test for question 3?
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WHY WAS THIS NOT RELEASED 4 DAYS AGO WHEN I DIDNT HAVE MY CALC 2 EXAM ON FRIDAY -purdue engineering student
Haha I really could have used this practice before I failed my calc final yesterday :)
I have mine today luckily
Are you able to give some information about dynamic systems? matrices etc. I have this question but unsure what to put as X1, Y1 and X2,Y2 "David jogs either in the woods or on asphalt. If he jogs in the woods, the probability is 30% that the next jogging trip is also in the forest, while the probability is 70% for the next jogging trip on asphalt.If he jogged on asphalt, the probability is 50% that the next trip is in the woods, and 50% the first trip on asphalt. David jogs his first trip in the woods. Find a formula for probability vector k, which describes the probabilities of forest or asphalt on trip k."
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Excellent video😎😎😎, a query could explain in a video or privately how it shows that this complex series ∑ (n/tan(i*π*n)), n=1 to infinity, is divergent
For question number 3 couldn't we use stirling's asymptotic formula too?
In question no 5 at 30:20 can we can also use ratio test using expansion of ln(x) and ln(x+1)?
I thought that if cos(0) is 1 and cos(1) is about 1/2 you would be adding closer and closer to one and the sum would be divergent.
Can you please make some vedio on 3-d geometry