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Current time:0:00Total duration:4:53

Divergent telescoping series

Video transcript

let's say that we have the sum 1 minus 1 plus 1 minus 1 plus 1 and just keeps going on and on and on like that forever and we can write that with Sigma notation this would be the sum from n is equal to 1 lower case N equals 1 to infinity we have an infinite number of terms here let's see this first one we want it to be a a positive 1 and then we could want to keep switching terms so we could say that this is negative 1 to the lowercase n minus 1 power let's just verify that that works when n is equal to 1 it's negative 1 to the 0 power which is that when n is equal to 2 it's 2 minus 1 it's negative 1 to the first power that's equal to that right over there so this is a way of writing this series now what I want to think about is does this series converge to an actual finite value or and this is another way of saying it's like what is the sum is that is there a finite sum that we can say is equal to this right over here or does this series diverge and the way that we can think about that is by thinking about it's partial sums let me write that down the partial the partial sums of this series and the way we could define the partial sums so we'll give an index here so capital n so the partial sum is going to be the sum from N equals 1 but not infinity but to capital n of negative 1 to the N minus 1 so just to be clear what this means so the the partial sum with just one term is just going to be from nth lowercase N equals 1 to upper case N equals 1 so it's just going to be this first term right over here it's just going to be 1 the S sub 2 S sub 2 is going to be equal to 1 minus 1 it's going to be the sum of the first two terms S sub 3 s sub 3 is going to be 1 minus one plus one it's the sum of the first three terms which is of course equal to equal to the C this is equal to one this one over here is equal to is equal to zero S sub four we could keep going S sub 4 is going to be 1 minus 1 plus 1 minus 1 which is equal to 0 again so once again the question is does this sum converge to some finite value and I encourage you to pause this video and think about it given what we see about the partial sums right over here so in order for a series to converge that means that the limit an infinite series to converge that means that the limit the limit so if you're convergence convergence is the same thing is the same thing as saying that the limit as capital as the limit as capital n approaches infinity of our partial sums is equal to some finite let me just write like this is equal to some finite finite value so what is this limit going to be well let's see if we can write this so this is going to be let's see S sub n if we want to write it in general terms we already see if if s if if Capital n is odd its equal to 1 if Capital n is even it's equal to 0 so we could write let's write this down so S sub n I could write it like this it's going to be 1 if n odd it's equal to 0 if n even so what's the limit as S sub n approaches infinity so what's the limit what's the limit as n approaches infinity of S sub n well this limit doesn't exist it keeps oscillating between these points you give me you go one more it goes from 1 to 0 you give me one more it goes from 0 to 1 so it actually is not approaching a finite value so this this right over here does not exist does not exist it's tempting because it's bounded it's all it's all it keeps oscillating between one and zero but it does not go to one particular value as n approaches infinity so here we would say that our series s diverges our series s die diverges