Absolute Convergence, Conditional Convergence and Divergence
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- Опубліковано 15 гру 2024
- In this video, we delve into the concepts of Absolute and Conditional Convergence using the Ratio Test. You will learn how to apply the Ratio Test to various series and understand whether they are absolutely or conditionally convergent or divergent.
We’ll walk through multiple examples, including:
Evaluating a series with factorial terms.
Using the Ratio Test to determine convergence.
Applying convergence conditions to alternating series with square roots and other functions.
By the end of the lesson, you'll have a strong grasp of how to tackle these types of problems using the Ratio Test and will be able to confidently determine the convergence of a series.
What You Will Learn:
How to apply the Ratio Test for determining convergence
How to differentiate between absolute and conditional convergence
Step-by-step breakdown of alternating series examples
Visual representations to aid understanding of complex series
Make sure to leave a comment if you find the video helpful, and don't forget to like, share, and subscribe! Also, let your friends, teachers, or parents know about this channel if you think they would benefit from these lessons.
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#RatioTest #Convergence #MathHelp #AbsoluteConvergence #ConditionalConvergence #Divergence #SeriesConvergence #PatrickJMT #Algebra #Calculus #FactorialSeries
you deserve 99% of my tuition
why not 100?
He needs the 1% for ramen
Your damn right! I started watching his video since freshman year in high school. Now I am in college and I still watch his video
patrickJMT: Allowing students to skip calc class since 2007
+TheYoungOne he is genius bro (Y)
Exactly patrick is awesome
Anytime I need to figure something out, you're my guy, Patrick. Thank you so much for all of your videos.
happy to help!
The way this guy writes his capital sigmas is pretty boss.
It's pretty horrendous. Writing Sigma is the one thing he can't seem to do.
This is gonna help me for my test, which is in like 5 mins
Alex Augustine #eng
+Alex Augustine mine is in 1 hour and 40 minutes
how did u do?
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are you joking?, or you did graduate? I know that's been a while.@@alibabagalaxy
i probably have said this a hundred times before, but i'll say it again, you're our saviour.
patrick man dnt knw whre would i be without u , since 2013 my first year at university of johannesburg, you the boss thanks, may success be with u through ur entire life
Thanks for passing me in Precalc, Calc I, Calc 2, and probably Diff. Eq next semester! You are the best!
How has life been 2 years later?
What about 3?
no calc 3?
One of the best ASMR's I've had the pleasure of having.
My Cal2 teacher in university cannot explain this. I use your videos to learn as substitute. It really helps. Thanks man.
Love how you do examples that are a bit challenging unlike the usual simple book examples
For anyone that's interested, the most interesting fact about conditionally convergent series (and the reason why we care at all whether or not a series is absolutely convergent) is this:
If a series is conditionally convergent, you can re-arrange the terms to make it converge to any number you wish.
The series 1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + ... converges to 0, but we can make it converge to 1 instead by doing the following:
(1) Add the positive terms until you get a number greater than 1
(2) Subtract the negative terms until you get a number less than 1
(3) Repeat (1) and (2) forever
1 + 1/2 - 1 + 1/3 + 1/4 - 1/2 + 1/5 + 1/6 + 1/7 - 1/3 + 1/8 + 1/9 + 1/10 - 1/4 + ...
This works because if the series is conditionally convergent then both the series consisting of the positive terms and the series consisting of the negative terms grow unbounded. So we never run out of positive and negative value to add. And the amount that we overshoot our target by each time converges to zero as the positive and negative terms converge to zero.
Okay, I've been learning from your videos for the past year or so, never actually needed to make a youtube account, just made up one, was sure to subscribe to the best channel on youtube. I always make sure to watch the ads to it's entire length because YOU DESERVE THE ULTIMATE RESPECT! Great Work Patrick! I hope to meet you in person one day and thank you for all your hard work! You're a great man Patrick!
Subscribed. I'm taking Calculus II this semester and I'm a non-traditional student. I may or may not major in math. I have been referencing your videos for the past year or so. Thanks for uploading.
this is your best video yet. midterm tomorrow and i recommended it to everyone i found in my hallway who was still awake!
At 1:28 I was about to scream out my typical classroom "WAIT don't erase that yet!" But then I remembered the amazingness of online lectures and rewind power.
Every time I think Patrick could not have possibly taken the time to cover sub part of something he does, Bravo sir! I am petitioning to have calculus renamed after you.
All three examples were the first three odd numbers in my book. This guys my hero!
Great work, really helping me get prepared and confident for my finals
Seriously man.. Im in calc2 summer classes and we get 1.5 weeks to go over series. You're a huge help.
@MustangGTR2 it gets used in differential equations, which is hugely important in physics, for example.
You were spot on when you said that more challenging examples are more useful. Please keep it up, it helps a lot!
@123Retry123 well, the ratio test uses absolute value so you are testing for absolute convergence at that point
@jvideogamer it is a geometric series; just rewrite so that it is all being raised to the n power, determine if the ratio is less than one (and greater than negative one)
Dude, you're awesome. i've been watching you since i first learned calculus in high school, and you've never failed me yet. Keep up the awesome job!!!
:D
one little piece of advice for you, just housekeeping, really, is to move the p-series videos BEFORE all the convergence tests, since you reference p-series before you explain them. it would clear a lot of things up for people like me watching this playlist from start to finish. :)
At the beginning the problem, Patrick used the absolute value to determine whether the series converges absolutely. As you saw on the video, by the p-series test it diverges. However, he used the Alternating test to check for conditional convergence. According to the Alternating Test, you do not need the (-1)^n but rather the rest of the series which is referred as the Bn. For that reason he did not use it.
Thank you, I was so confused 🙏🏽
patrickJMT is the freaking MAN
Listening to my Calc II professor drone on for two hours makes my eyes bleed, but listening to Patrick for 10 minutes is like cuddling with 100 puppies, while actually understanding concepts.
I switched from engineering but I finished the math curriculum in my school and his videos went all they way up to Differential Equations/Linear Algebra.
patrick jmt is like my savior for calculus
holy crap how does he make it so easy to understand? I am blown away at his method of teaching
You are a deity to me.
@PolarisUSMC i used to teach in a classroom. i still take 'private students' (that is, i do a bit of tutoring). however, i do enjoy making the videos and chatting with random people out there on the internets, so it is also 'for fun'
The terms "conditional convergence" and "absolute convergence" were really confusing, but the very beginning of your video answers my question. The rest of the process is relatively simple. Thank you!
@260191894 but math majors in your class do need to know how to prove theorems, and they are not gonna hear it in chemistry class : )
glad you like the videos!
i'm probably going to end up watching all of your videos before my final exam.
thank you so much!
@xcaliberpeng it is inconclusive
The fact that this is actually making any sense is insane! But thanks Patrick, ur the best
The world wouldn't be the same without you!
You have helped me sooo much in Calc 2 and Calc 3. Keep up the good work! Keep those videos coming!!
I keep using this video as a reference to my work... You're the man!
Because the bottom approaches infinity and the top stays at 10 which means that the bottom is WAYYYYY larger than the top. Think of it like saying 1/8, 1/9, 1/10, 1/11, etc -- every time the bottom gets bigger and the top says the same (or at least less than the top), it (the fraction) gets closer and closer to 0! Hope that helps...
You and doc schuster are real love of my life!
Thanks for all your videos, I survived first and second year math/calculus because of your videos thanks a lot!!
thanks dude! You're this generation's hero!
@iniloy1993 you do whatever you feel is necessary with your like button : )
Thanks a lot Patrick! From Honors Pre Calc to Calc BC you've always been my go to guy whenever I get lost in some of the challenging concepts. More power!
Hey man really helpful video.. just one doubt @9:35 which test do you refer to?
Patrick is the man! Thanks a bunch bro. My collage profs can't even explain like you do!!!!
this video taught me more than my calc prof did in a whole semester
good stuff man! all the way from australia, first year engineering maths has never been easier, i havent showed up to a single math lecture this semester since i found your videos!
you are the reason I pass calc 3 this semester! Thanks so much!
Haven't you heard? He's so pro at math that he got bored and started moving on to other subjects (he just started teaching himself physics a couple months ago).
@MrBrandonthegiant tryin' to keep it neat for all you people out there
yes
You're making a lot of lives a lot easier ya know, thanks
You do a better job at teaching than my calculus teacher. 👍
(n+1)! = (n+1) * (n)! = (n+1) * (n) * (n-1)! ...
or if we substitute in numbers... (let n = 8)
9! = 9 * 8! = 9 * 8 * 7! ...
(in the problem he does (n +1)! = (n + 1) * n! and then cancels out the n! because it also appears on the bottom, leaving (n + 1).
Hope I helped! Once it clicks you'll get it, substituting in real values helped me realize that (n + 1) is just an integer & nothing more complex than that.
Very good well done, only a leftie could explain with this level of awesomeness. thanks patrick
For the conditionally convergent test for problem 2 and 3, you said on 2, that its an alternating series, so you ignore the alternating part (-1)^n+1, but why didn't you ignore it for 3?
You made it look so easy, and quite frankly, it is... I finally get it, thanks! :-) This makes more sense than my thick Calculus book..
So does conditional convergence basically mean the convergence or divergence for the absolute value of the function and the regular function are different? Does it matter which one of the two is convergent or divergent? The way you explained it it seems like it ONLY will occur when (an) converges and |(an)| diverges, not the other way around.
I went from failing calc 2 to having an 84% right now, partially thanks to you!
THIS MAN IS A GOD.
This lecture is really conceptual and secured my marks...Good Job
@mshiraa b/c the equation alternates between +1 and -1. It doesn't fit the requirements that equation (b sub n ) is eventually decreasing. Technically it is diverging, (that's what you would put on the paper if this came up) but the real answer is it doesn't exist, because you don't know what your final value is, since infinity is neither odd or even.
Great addition to the Series videos! I totally get it now!
patrick love you!!!! saving me in my exams
The last one should be inconclusive right? Although the original bsubn diverges after doing the absolute convergence test the limit as n approaches infinity ends up being 1, therefore yielding an inconclusive answer for convergence/divergence.
thanks patrick, you are the best !!!
Keep up the good work Patrick! You're the greatest :)
my only calculus quiz went well just because of you
for the last example can you use the alternating series test to prove that it diverges? since the limit does not equal 0 and it's not a decreasing sequence due to the first derivative test
When checking for Conditional convergence you don't consider the (-1)^n part. So the last step should be { lim (n/n+5) }.
In fact if the series is not absolutely convergent by using Divergence ,Ratio or Root test you don't have to check if it's conditionally convergent since it will always be divergent.
@5otrebor5 yes
hey shouldn't you ignore the (-1)^n part when ur finding the limit for last problem???? that way you get limit =1 which is not 0, and therefore diverges?????????
When you use the ratio test and the test proves it is divergent, does it mean the whole series is divergent or can you check to see if it's conditionally convergent?
Thanks in advance.
Four words……Thank GOD for patrick
Thanks Patrick, your videos are really helpful. keep up the good work. You help us getting A+ on the exam
Just wanted you to know that you and your videos are a godsend :D
This was remarkable. Straight to the point. Thank you
Literally saving my grades , Thanks man
Why did you include (-1)^n on your last example when testing for conditional convergence? I thought I understood that the limit would be computed without that factor. Either way, you reach the same answer. Am I wrong?
May Allah grant you Jannah and inshallah make your life easier for you. بارك الله فيك
Thank you so very much! You have no idea how much I needed this.
Thanks again!
In the 3rd example the lim equaled to 1 so by ratio test doesnt it mean that no conclusion can be drawn about the convergence or divergence of that series as stated in the book?
For the alternating series n/(n+5), is it possible to determine conditional convergence by simply showing that the series increases? I know PatrickJMT was able to show that the limit approaches -1 and +1, thus showing that the limit DNE, so I am only asking about checking for bn > bn+1 as a viable alternative
Thank you patrick for the more difficult questions. More often than not I find questions that are too simple for my schools level..
Once again, you have saved my grade :)
Engineers don't have to know about the derivations of the equations and relationships and write up proofs to show that the equations hold. A lot becomes abstract and it's not so much of "here's a formula/algorithm and this is how to use it."
7:20 u're using leibniz (-1)^n X a^n u see if A^n as u said and also limit to 0 then it's a CV
In your second and third examples, when testing for conditional convergence, in the second example you say to neglect the (-1)^n but in the third you include it. Por que?
in the second example, when testing for conditional convergence, you neglected the term that makes it alternate and took the limit. Why did you not do the same when testing for conditional convergence in the last example?
since the 2nd example was conditionally conv, do you also have to say its cond conv by the alt series test or no?
You explain everything so clearly. Thank you very much.
Could there be a case where the absolute value converges but the regular alternating series diverges? I would guess no, but I can't articulate why.
We said we can neglect the alternating sign part but on problem 3 why did you count it for solving original series?
How come you didn't use the alternating series test when you were trying to find if the series was conditionally convergent in 10:02?
Hello Patrick, first of all, thank you for all your videos, they are very much appreciated.
I have a question regarding the conditional convergence test. In the second example, when you test for conditional convergence, you leave out the alternating part of the series to test for the limit, but in the third example, you use a(sub n) with the alternating part. Could you please explain why that is? When should I leave out the (-1)^(n+1) ?
Again, thank you!