Absolute Convergence, Conditional Convergence and Divergence

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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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What about 3?

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no calc 3?

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Love how you do examples that are a bit challenging unlike the usual simple book examples

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@MustangGTR2 it gets used in differential equations, which is hugely important in physics, for example.

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My Cal2 teacher in university cannot explain this. I use your videos to learn as substitute. It really helps. Thanks man.

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You were spot on when you said that more challenging examples are more useful. Please keep it up, it helps a lot!

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All three examples were the first three odd numbers in my book. This guys my hero!

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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)

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Very nice and clear explanation! Thank you very much!

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this was an awesome video. Cleared things up for me so well. Thank you

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@123Retry123 well, the ratio test uses absolute value so you are testing for absolute convergence at that point

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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.

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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'

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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!

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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!

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[@donyajohari6756]

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