Absolute Convergence, Conditional Convergence and Divergence
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- Опубліковано 21 кві 2009
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Absolute Convergence, Conditional Convergence and Divergence for series. In this video, I give the basic result and do 3 examples!
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This is gonna help me for my test, which is in like 5 mins
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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.
You were spot on when you said that more challenging examples are more useful. Please keep it up, it helps a lot!
@MustangGTR2 it gets used in differential equations, which is hugely important in physics, for example.
Love how you do examples that are a bit challenging unlike the usual simple book examples
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@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)
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.
@123Retry123 well, the ratio test uses absolute value so you are testing for absolute convergence at that point
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Seriously man.. Im in calc2 summer classes and we get 1.5 weeks to go over series. You're a huge help.
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All three examples were the first three odd numbers in my book. This guys my hero!
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@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'
Patrick is the man! Thanks a bunch bro. My collage profs can't even explain like you do!!!!
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!
Thank you so very much! You have no idea how much I needed this.
Thanks again!
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Great addition to the Series videos! I totally get it now!
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this was an awesome video. Cleared things up for me so well. Thank you
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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.
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Very nice and clear explanation! Thank you very much!
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This lecture is really conceptual and secured my marks...Good Job
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Great stuff! Really helped for my AP studying.
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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!
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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 🙏🏽
this video taught me more than my calc prof did in a whole semester
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..
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@xcaliberpeng it is inconclusive
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thank you so much!!! u r way better to explain this than my teacher :D
Very good well done, only a leftie could explain with this level of awesomeness. thanks patrick
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Hey man really helpful video.. just one doubt @9:35 which test do you refer to?
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@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!
This is really helps thanks! just in time for my test tomorrow >.
Great work sir..
you can use the alternating series for the last problem as well, for series an.
I freaking appreciate you.
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. :)
Thank you Patrick :) good stuff
thanks mahn ..appreaciate it
helped a lot
Thanks for the help.