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# Conditional & absolute convergence

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.12 (EK)
,
LIM‑7.A.13 (EK)

## Video transcript

in the video where we introduced the alternating series test we in fact used the series we use the infinite series from N equals one to infinity of negative 1 to the n plus 1 over N we use we use this as our example to apply the alternating series test and we prove that this thing right over here converges so this series which is 1 which is 1 minus 1/2 plus 1/3 minus 1/4 and it just keeps going on and on and on forever we use the alternating series test in that video to prove that it converges so this thing converges so this converges converges by alternating series test by alternating alternating series alternating Limi might then looks a little messy alternating series test and if you want to review that go watch the video on the alternating series test now let's take a little bit about what happens if we were to take the absolute value of each of these terms so if you were to take the absolute value of each of these terms so if you take the sum from N equals 1 to infinity of the absolute value of negative 1 to the n plus 1 over N well what is this going to be equal to well this numerator is going to be 1 or negative 1 the absolute value that's always going to be 1 so it's going to be that over and n is always positive we're going from 1 to infinity so it's just going to be equal to the sum it's going to be equal to the sum from N equals 1 to infinity of 1 over N and this is just the famous harmonic series and there's this video that we have then you should look it up on Khan Academy if you don't believe me on the famous proof that the harmonic series diverges so the harmonic series is 1 plus 1/2 plus 1/3 this thing right over here this thing right over here the dye diverges and so when you see a series that converges but if you were to take the absolute value of each of its term and then that diverges we say that this series converges conditionally converges you can say it converges but you could also say it converges conditionally conditionally and the condition is I guess you could say that we're not taking the absolute value of each of the terms and the if it if something converges when you take the absolute value as well then you say it converges absolutely and so let's look at an example of that if I were to take if I were to take if I were to take this series let's do a geometric series that might be fun actually I'm using these colors too much let me use another color let's say let's take the sum from N equals 1 to infinity of negative 1/2 to the n plus 1 power we know this is a geometric series where the absolute value of our common ratio is less than 1 we know that this converges we know that this converges and if we were to take the absolute value of each of these terms so if you were to take the sum let me do that in a different color just to mix things up a little bit if you were to take the absolute value of each of these terms so the absolute value of negative 1/2 absolute value of negative 1/2 to the n plus 1 power this is going to be the same thing as the sum from N equals 1 to infinity of 1/2 to the n plus 1 and here once again the common ratio the absolute value of the common ratio is less than 1 and we studied this when we looked at geometric series this also converges this also converges so when we took the absolute value of the terms is still converged so for this one we can say that this converges absolutely absolutely so we've talked a lot already about convergence or divergence and that's all been good and what we're doing in this video we're introducing a nuance or flavors of convergence so you can converge but it might be interesting to say well do we convert would it still converge if we took the absolute value of the terms if it won't if you converge but it doesn't converge when you take the absolute value of the terms then you say it converges conditionally if it converges and it still converges when you take the absolute value of the terms then we say it converges absolutely because even if you take the absolute value of the terms it converges hopefully you find that interesting
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